Stochastic modelling of insulin sensitivity variability in critical care

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Biomedical Signal Processing and Control 1 (2006) 229–242

Stochastic modelling of insulin sensitivity variability in critical care

Jessica Lin a,*, Dominic Lee b, J. Geoffrey Chase a, Geoffrey M. Shaw c,d,Christopher E. Hann a, Thomas Lotz a, Jason Wong a

a Department of Mechanical Engineering, Centre for Bio-Engineering, University of Canterbury, Private Bag 4800, Christchurch, New Zealandb Department of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch, New Zealand

c University of Otago, Christchurch School of Medicine and Health Sciences, Christchurch, New Zealandd Canterbury District Health Board, Department of Intensive Care Medicine, Christchurch Hospital, New Zealand

Received 9 February 2006; received in revised form 12 September 2006; accepted 12 September 2006

Available online 27 December 2006

Abstract

Tight glycemic control has been shown to reduce mortality by 29–45% in critical care. Targeted glycemic control in critical care patients can be

achieved by frequent fitting and prediction of a patient’s modelled insulin sensitivity index, SI. This parameter can vary significantly in the critically

ill due to the evolution of their condition and drug therapy.

A three-dimensional stochastic model of SI variability is constructed using 18 long-term retrospective critical care patients’ data. Given SI for an

hour, the stochastic model returns the probability distribution of SI for the next hour. Consequently, the resulting glycemic distribution 1 h

following a known insulin and/or nutrition intervention can be derived. Knowledge of this distribution enables more accurate predictions for

glycemic control with pre-determined likelihood based on confidence intervals.

Clinical control data from eight independent critical care glycemic control trials were re-evaluated using the stochastic model. The stochastic

model successfully captures the identified SI variation trend, accounting for 84% of measurements over time within the 0.90 confidence band, and

45% with a 0.50 confidence. Incorporating the stochastic model into the numerical glucose–insulin dynamics model, a virtual cohort was

generated, imitating typical glucose–insulin dynamics in a critically ill population. Control trial simulations on this virtual cohort showed that the

0.90 confidence intervals cover 88% of measurements, and the 0.5 confidence intervals cover 46%. These results indicate that the stochastic model

provides first order estimate of insulin sensitivity, SI, variation and resulting glycemic variation in critical care.

# 2006 Elsevier Ltd. All rights reserved.

Keywords: Stochastic Markov modelling; Insulin sensitivity; Blood glucose; Intensive care; Adaptive control

1. Introduction

Critically ill patients often experience stress-induced

hyperglycemia and high levels of insulin resistance, even

given no history of diabetes [1–10,35]. The metabolic response

to stress is characterised by major, highly variable changes in

glucose metabolism. Increased secretion of counter-regulatory

hormones leads to a rise in endogenously produced glucose and

the rate of hepatic gluconeogenesis, and a concomitant

reduction in insulin sensitivity. Tight glucose control has been

shown to reduce intensive care unit (ICU) patient mortality by

* Corresponding author. Tel.: +64 3 364 2987.

E-mail addresses: [email protected] (J. Lin),

[email protected] (J.G. Chase).

1746-8094/$ – see front matter # 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.bspc.2006.09.003

45% if average glucose is less than 6.1 mmol/L for a cardiac

care population [9,10]. Krinsley [12] showed a 25–30% total

reduction in mortality over a broader critical care population

with a higher average glucose limit of 7.75 mmol/L. Therefore,

control algorithms that provide tight regulation for glucose

intolerant ICU patients would reduce mortality and the burden

on time and medical resources.

Previous clinical glycemic control studies include [13–18].

Chase et al. [13] and Doran et al. [16] focused on critical care

patients, whose glucose–insulin dynamics are highly variable

due to the stress of their illness and the impact of drug therapies.

Chase et al. [13] developed a control algorithm that has been

clinically verified in ICU to reduce elevated blood glucose

levels in a controlled, predictable manner, while accounting for

inter-patient variability and varying physiological condition.

The overall approach is a targeted adaptive control scheme that

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http://dx.doi.org/10.1016/j.bspc.2006.09.003

J. Lin et al. / Biomedical Signal Processing and Control 1 (2006) 229–242230

identifies changes in patient dynamics, particularly with respect

to insulin sensitivity.

Following Chase et al. [13], better understanding and

modelling of patient variability in the critical care population

can lead to better glycemic management in ICU. In particular, a

common risk in any intensive insulin therapy is hypoglycemic

events. Many current ad hoc intensive insulin therapy protocols

have reported hypoglycaemic episodes up to 25% of incident

rate [9,19–21]. Many studies have shown that an episode of

hypoglycaemia can lead to counter-regulation and severe

rebound hyperglycemia, which is particularly difficult to

control [22–25]. Hypoglycaemic episodes therefore present a

significant added risk in providing intensive insulin therapy in

the ICU.

Understanding and modelling the variability in patient

condition, or more specifically, the patients’ variable dynamic

response to insulin, will thus assist clinical control intervention

decision making, and minimise the associated risk. Currently,

no intensive insulin therapy protocol offers the likelihood, or

distribution, of glycemic response to an intervention, leaving

clinicians partly blind in controlling such a highly dynamic

system. Therefore, the ultimate goal of this study is to produce

blood glucose distributions and confidence bands for control

intervention decisions based on stochastic models of clinically

observed parameter variations. Such bands will allow targeted

control, with user specified confidence on the glycemic

response outcome. The result will be added certainty and

safety in providing tight glycemic control.

2. Glucose–insulin system model and parameter

identification

Tight glucose control requires a patient-specific model that

captures the fundamental dynamic responses to elevated

glycemic levels and insulin. The model used in this study is

a patient-specific two-compartment glucose–insulin system

model from Chase et al. [13] and Hann et al. [26]. The

physiologically verified model accounts for time-varying

insulin sensitivity and endogenous glucose removal, along

with two different saturation kinetics.

2.1. Glucose–insulin system model

The glucose–insulin system model is presented in Eqs. (1)–

(3):

G ¼ � pGG� SIðGþ GEÞQ

1þ aGQþ PðtÞ (1)

Q ¼ �kQþ kI (2)

I ¼ �nI

1þ aIIþ uexðtÞ

V I

(3)

where G and I denote glucose above an equilibrium level, GE,

and the plasma insulin from an exogenous insulin input,

respectively. Insulin utilisation over time is captured by Q,

where k is the effective insulin half-life parameter. Patient

endogenous glucose removal and insulin sensitivity are pG

and SI, respectively. The parameter VI is the insulin distribution

volume and n is the first order decay rate for plasma insulin.

External nutrition and insulin inputs are P(t) and uex(t), respec-

tively. Michaelis–Menten parameters aI and aG define plasma

insulin disappearance saturation and insulin-stimulated glucose

removal saturation, respectively.

Insulin sensitivity is the critical parameter that drives the

observed dynamics of the system for critical care patients [26].

Therefore, accurate identification of insulin sensitivity is an

important part of any glycemic control protocol in the highly

variable critical care population. Methods for determining

insulin sensitivity have been extensively studied and are highly

dependent on the experimental protocol and/or dynamic model

adopted [e.g. 27,28–30]. Hyperinsulinemic euglycemic clamp

tests with different levels of plasma insulin concentration are

the gold standard, but also give very different insulin sensitivity

levels depending on dose and intra-individual variation [27,31].

In Eq. (1), the saturation mechanism on insulin effect creates

a unique index of insulin sensitivity, SI, compared to other

model-based measures. The result is an SI index that more

closely approximates the tissue sensitivity to insulin, and its

variation to the evolution of patient condition and drug therapy.

This model measure is also highly correlated to clamp-derived

ISI over 146 patients [32,33]. Identifying this parameter, and its

variation over time as patient condition evolves, are thus critical

to providing safe, effective tight glycemic control. Equally

important, a model of the stochastic variation in insulin

sensitivity would enable development of much more robust

likelihood-based glycemic control protocols, given this added

knowledge.

2.2. Integral-based parameter identification

Using constant population values for aG, aI, n, k and VI limits

the model unknowns to pG and SI. This study utilises an

integration-based parameter identification method first pre-

sented in Hann et al. [26] for accurate identification of pG and

SI. A thorough Monte Carlo sensitivity study of the other

parameters was performed in Hann et al. [26], reinforcing the

method of simplifying the problem and limiting unknowns to

the two crucial variables. Wong et al. [34,35] achieved 94% of

predictive control accuracy by fitting SI alone. In reality, n and k

associated with transport and utilisation rate of insulin cannot

be clinically measured in real time, and therefore need to be

assigned population values for the model to be applied in real-

time control. There exist tradeoffs between fitted SI and

assumed aG [13,36]. However aG is difficult to be identified,

therefore a value that has been tested to give the best fitting

result overall is assumed for the entire retrospective cohort, and

produced an average fitting error of 4.3% (0.87–7.42%) [26].

The functions defining pG and SI can be arbitrarily designed.

The integral fitting method results in a convex least squares

problem that demands little computational time and intensity.

In contrast, the commonly used non-linear recursive least

squares routine is non-convex and starting point-dependent [37]

and can thus deliver inaccurate results [38]. The integral

Table 1

Constant population parameter values

Parameter Unit Value

aG L/mU 1/65

aI L/mU 0.0017

n min�1 0.16

k min�1 0.0198

VI L 3.15

Fig. 1. Fitted hourly SI variation and probability distribution function.

J. Lin et al. / Biomedical Signal Processing and Control 1 (2006) 229–242 231

approach is also robust to measurement noise, as it effectively

low-pass filters the data. Finally, constraints are placed on both

parameters to ensure they are within physiologically valid

ranges. Details of the fitting routine and population values

utilised are outlined in Hann et al. [26].

3. Stochastic modelling

The control algorithm of Chase et al. [13] calculates the

interventions necessary for targeted glycemic regulation by

assuming that the identified pG and SI values are constant

between the control intervention and the 1-h time interval to a

pre-selected target. However, identified profiles of pG and SI

have shown that both variables evolve significantly through

time based on patient condition [26]. In particular, sudden

variations may also occur due to onset of conditions such as

atrial fibrillation [39]. A verified model of the variability of pG

and SI would thus significantly assist clinical control

intervention decision making.

The main idea is to provide information to enable the control

system to minimise the risk of unexpected glycemic excursion,

particularly hypoglycemia. The ultimate goal of this stochastic

modelling is to produce blood glucose confidence bands based

on clinically observed parameter variations for a given

intervention. Such bands would allow glycemic target selection

with guaranteed levels of certainty that a result meet or exceed a

given glycemic level based on an empirical model created from

clinical data. Thus, such a model would add safety and better

knowledge to glycemic control decision making.

3.1. Stochastic parameter model

Patient-specific parameters, pG and SI are fitted to long-term

retrospective clinical data from 18 critical care patients in the

Department of Intensive Care Medicine, Christchurch Hospital.

These 18 patients are a selection from a 201-patient data audit

at the Christchurch Hospital [26,40,41]. Each patient record

had a period greater than 1 day with intervals between

measured data points of 3 h or less. This cohort broadly

represents a typical cross section of ICU patients, regarding

medical condition, age, sex, APACHE II scores and mortality.

Diagnosed Type 1 and Type 2 diabetes are slightly over-

represented because they often received greater monitoring.

Zero-order piecewise linear functions are used to define the

potential variability in the modelled pG and SI, with a

discontinuity every 2 h for pG and every hour for SI. Greater

variability is given to SI because studies have shown that it is

much more variable than pG [26], matching physiological

expectations. A summary of the population parameter values

used is shown in Table 1, and are based on an extensive

literature search [e.g. 31,42,43–45].

The fitted pG and SI profiles from the 18 long-term critical

care patients reveal non-uniform variation patterns with respect

to the parameter values themselves. More specifically, the

variability of both parameters over any given hour is dependent

on its present value, and that the stochastic behaviour or

distribution of these variations depends on their current state. A

two-dimensional kernel density estimation method is used to

construct the stochastic model that describes the transition of

parameter values from 1 h to the next, with respect to the

parameter values. The method has the advantage of producing a

smooth, continuous function across the parameter range

[46,47]. The overall result is a bivariate probability function

for the potential parameter values.

The variation distribution of fitted SI from the 18 patients,

plotted as SI n+1 against SI n, is shown by the dots in Fig. 1. The

contours of the probability function for potential SI are also

shown in Fig. 1, as shaded areas. As can be seen from Fig. 1, the

probability function peaks at where there is the highest data

density.

Essentially, kernel density estimation methods enable data

extrapolation to the entire population given this type of sample

from the population. The two-dimensional kernel estimation

method provides an approximation to the parameter variation

behaviour according to how the existing data behaves. Where

there is higher density of data, more certainty can be drawn on

the ‘‘true’’ behavioural pattern of the variant.

An example of a visually and conceptually simpler one-

dimensional kernel density estimation method is shown in

Fig. 2. The kernel density estimate r(x) is the solid line and the

kernel functions that add up to r(x) are dashed. Six sample

points were considered. Note that where the points are denser,

the density estimate has higher values. Hence, these

approximations are more certain and thus more accurate in

the presence of significant data points. In contrast, lack of data

Fig. 2. One-dimensional kernel density estimation (the kernel density estimate

r(x) is the solid line; the kernel functions which add up to r(x) are dashed).


cannot be remedied by this approximation, as might be

expected.

On the same principle, the two-dimensional kernel density

estimation process can be thought of as a sand building exercise

for visualisation purposes. If a pile of sand is dropped onto

every data point dots in Fig. 1, then the resulted sand sculpture

is the simple representation of the kernel density estimate

across the SI n � SI n+1 plane, much like the solid line in Fig. 2.

However, the sand sculpture is physically constrained to the

positive first quadrant. Thus, the probability of non-positive,

physiologically invalid SI values is eliminated with this added

boundary on the density estimate.

3.2. Markov model

Since the probability distribution of a possible future SI n+1

value depends on SI n, time-varying SI can be treated as a

Markov chain. A Markov chain constitutes a sequence of

random variables, X0, X1, X2, X3, . . ., with the value Xn being the

state of the process at time n. The Markov property states that

the conditional probability distribution of future states of the

process, given the present state, depends only upon the current

state. Therefore, the conditional probability distribution of Xn+1

given past states is a function of Xn alone:

PðXnþ1jX0;X1;X2; :::;XnÞ ¼ PðXnþ1jXnÞ (4)

Additionally, the conditional probability has the statistical

property:

PðAjBÞ ¼ PðA;BÞPðBÞ (5)

Therefore, given the Markovian stochastic behaviour of SI, the

conditional probability of SI n+1 taking on a value y can be

calculated by knowing or having identified SI n = x:

PðSI nþ1 ¼ yjSI n ¼ xÞ ¼ PðSI n ¼ x; SI nþ1 ¼ yÞPðSI n ¼ xÞ (6)

Eq. (6) is the conditional probability function that will provide

the stochastic information needed on potential SI variation. The

numerator on the right hand side, which is the two-dimensional

kernel density estimated joint probability P(x, y), is constructed

upon the available data:

Pðx; yÞ ¼ 1

n

Xn

i¼1

fðx; xi; s2xiÞ

pxi

fðy; yi; s2yiÞ

pyi

(7)

where

pxi¼Z 1

0

fðx; xi; s2xiÞ (8)

pyi¼Z 1

0

fðy; yi; s2yiÞ (9)

and xi and yi are the coordinates of each dot in Fig. 1. Eq. (7) is

the two-dimensional kernel density estimator function. Each

fðx; xi; s2xiÞ and fðy; yi; s

2yiÞ is a normal probability distribution

function centred at corresponding xi and yi. To force non-

negativity in x and y, Eqs. (8) and (9) provide normalisation

in the positive domain, where each pxiand pyi

represents the

area under each fðx; xi; s2xiÞ and fðy; yi; s

2yiÞ between zero and

infinity.

Since

PðBÞ ¼Z

PðA;BÞ dA (10)

The denominator on the right-hand side of Eq. (6) can be

calculated by integrating Eq. (7):

PðxÞ ¼Z

Pðx; yÞ dy ¼ 1

n

Xn

i¼1

fðx; xi; s2xiÞ

pxi

Zfðy; yi; s

2yiÞ

pyi

dy

¼ 1

n

Xn

i¼1

fðx; xi; s2xiÞ

pxi

� 1

(11)

Therefore, Eq. (6) can be calculated from Eqs. (7) and (11):

PðSI nþ1 ¼ yjSI n ¼ xÞ

¼Pn

i�1ðfðx; xi; s2xiÞ= pxi

Þðfðy; yi; s2yiÞ= pyi

ÞPnj¼1 fðx; x j; s2

x jÞ= px j

¼Xn

i¼1

viðxÞfðy; yi; s

2yiÞ

pyi

(12)

where

viðxÞ ¼fðx; xi; s

2xiÞ= pxiPn

j¼1 fðx; x j; s2x jÞ= pxi

(13)

In conclusion, Eqs. (12) and (13) define the two-dimensional

kernel density estimation in conditional SI variability. Note that

SI variability is ‘‘conditional’’ because it depends on the prior

state of SI. Knowing SI at any hour n, SI n = x, the probability of

SI at hour n + 1, SI n+1 = y, can be calculated from Eq. (12).

3.3. Algorithm

The step-by-step description for how P(SI n+1 = yjSI n = x) is

computed is performed is as follows:

(1) F
or every fitted SI data point, shown by the dots in Fig. 1 and
identified as (xi, yi) where i = 1, 2, . . ., n, calculate pxiand

pyiusing Eqs. (8) and (9). The variance sxi and syi

at each

(xi, yi) depends on the local data density and is calculated


directly. Details of the derivation of s are shown in

Appendix A.

(2) K
nowing or identifying the value SI n = x, calculate vi(x) for
i = 1, 2, . . ., n using Eq. (13).

(3) C
alculate the probability that SI n+1 = y given SI n = x,
P(SI n+1 = yjSI n = x), by substituting vi(x) for i = 1, 2, . . ., n

into Eq. (12).

Step 1 only needs to be carried out only once, because it

depends solely on the existing data set used to construct the

stochastic model. The calculated pxiand pyi

can then be stored

for use in steps 2 and 3.

A better illustration of the construction of the (conditional)

stochastic model can be shown by the following example using

a data set of only eight samples for simplicity, as shown in

Fig. 3. Example of two-dimension

Fig. 3. Panel A shows the eight data points and the contours of

the individual kernels on them. Each kernel is a two-

dimensional Gaussian skewed by a weighting function vi(x)

as defined in Eq. (13). The weighting function skews the

Gaussian kernels in the x-direction with respect to the x-axis

data density at each data point, as shown in panel B. The overall

kernel density estimation function is then the sum of the

individual kernels as defined in Eq. (12), shown in panel C.

In summary, the two-dimensional kernel density estimation

method creates a smooth, continuous model surface that reflects

the sample data pattern. Note that the example shown is the

‘‘conditional’’ two-dimensional kernel density estimate func-

tion as defined in Eq. (12). Every slice of the surface in panel C

along the y-axis is the probability distribution in y (SI n+1) given

x (SI n), and therefore its area under the curve along the y-axis

al kernel estimation function.

Fig. 4. Three-dimensional stochastic model of SI variability.


sums to 1.0. In comparison, the kernel density estimation joint

probability function defined in Eq. (7) has the volume under the

three-dimensional surface equal to 1.0. The final three-

dimensional SI stochastic model is thus developed and shown

in Fig. 4 for this study.

3.4. Simple clinical implementation method

Having constructed the SI stochastic model, a grid of data

that describes the surface shown in Fig. 4 can be stored and used

as a look-up table. Having an identified hourly SI value in

clinical situations [13,35,39], the probability distribution, and

Fig. 5. Probability distrib

hence the confidence bounds, can be gathered, as shown in

Fig. 5. The solid line is the kernel density estimate surface

sliced along SI n = 0.6 � 10�3. This line represents the

probability distribution for potential SI n+1, 1 h after having

identified the current hour SI n = 0.6 � 10�3. From this

distribution, probability bounds are also obtained, giving the

most likely SI value in an hour at 0.58 � 10�3, inter-quartile

range [0.51 � 10�3, 0.65 � 10�3], and the 0.90 probability

interval [0.39 � 10�3, 0.75 � 10�3]. This probabilistic infor-

mation can then be used to assist the assessment of the patient

condition and clinical decision making regarding the optimal

intervention over the next hour.

ution of potential SI.

Fig. 6. Distribution of changes in pG.


3.5. Stochastic model for pG

The same kernel density estimation operations described

also applies to the endogenous glucose removal parameter pG.

However, the probability density across the x–y plane is highly

concentrated along the line y = x, and particularly at point

[0.01, 0.01], as shown by the scatter plot in Fig. 6. This result

reinforces the result that endogenous clearance, as modelled

by pG, generally stays constant at a patient-specific value [26].

From Hann et al. [26], its variation, when it occurs, is over

days rather than hours, and thus not as amenable to this

approach.

Hence, the variability of pG is neglected in this study. In

addition, calculating the joint probability distribution

between pG and SI requires significantly more computational

effort and time than considering SI alone for potentially little

gain in this instance. Finally, as this research is focused

on clinical glycemic control, computational simplicity is

also essential in permitting fast real-time clinical control

interventions.

3.6. Validation using clinical control data

The stochastic parameter model can be integrated into the

glucose–insulin system model of Eqs. (1)–(3). This step allows

the blood glucose level probability distribution 1 h following a

known insulin [13] and/or nutrition [35] intervention to be

defined based on the defined distribution of SI n+1. The

stochastic model therefore enables more knowledgeable

predictions with defined probability distributions for the

glycemic outcomes of glycemic control interventions.

The stochastic model developed from the 18-patient cohort

in Hann et al. [26] is evaluated on eight previous clinical control

trials in the ICU [13]. Importantly, these trial data are

independent from the 18-patient cohort used to develop the

stochastic model. The trials performed consist of hourly cycles

of the following steps [13,35]:

(1) M
easure blood glucose levels.
(2) F
it pG and SI to the measure blood glucose levels according
to past insulin and/or nutrition inputs using integral-based

parameter identification.

(3) D
etermine new control intervention to achieve the desired
blood glucose level using the identified current pG and SI.

(4) I
mplement control intervention.
To assess the stochastic model developed on its clinical

control validity, these eight trials are numerically performed

using the following modified cycle of steps:

(1) F
it pG and SI to an hourly cycle of blood glucose data using
integral-based parameter identification. (Control inputs are

as given in the clinical trial.)

(2) G
enerate probability intervals of potential SI from the time-
average identified SI of the evaluated cycle using the

stochastic model developed for SI.

(3) C
alculate blood glucose confidence intervals with respect to
the SI probability intervals using the numerical model

presented in Eqs. (1)–(3).

(4) C
ompare blood glucose confidence intervals to real blood
glucose trial measurements.

3.7. Further model applications: virtual patients and trial

simulations

Incorporating the stochastic model into the glucose–insulin

system model presented in Eqs. (1)–(3), typical critical care

patient dynamics can be reproduced numerically given an

initial SI value. Initial SI values are randomly produced from the

available retrospective SI data. Profiles of SI that are

representative of ICU patient condition evolution are generated

from the stochastic parameter model. Adapting these profiles

into Eqs. (1)–(3), ‘‘virtual patients’’ are created. A virtual

control trial can thus consist of hourly cycles of the following

steps:

(1) G
enerate hourly pG and SI values from the stochastic
parameter model probability distributions.

(2) G
enerate virtual blood glucose levels using generated pG
and SI values and specified control interventions in

Eqs. (1)–(3).

(3) F
it pG and SI to generated blood glucose levels using
integral-based parameter identification.

(4) G
enerate probability intervals of potential SI from the time-
average identified SI obtained from 3, using the stochastic

model developed for SI.

(5) S
pecify control intervention that produces the most
desirable probability distribution of potential blood glucose

levels.

Control interventions, in this case, include insulin bolus

injections, insulin infusions, and dextrose infusions, with

insulin injections being the primary resort as per Wong et al.

[35].

Using the stochastic model, calculated control inputs always

limit the lower 0.90 confidence interval bound to be above a set


hypoglycaemic limit. Hence, at high glycemic levels, control

interventions are calculated using these models to produce the

greatest likelihood at the target glucose level. However, once

the glycemic levels are approaching the normoglycemic range

of 4–6 mmol/L, the control interventions are designed to result

in blood glucose levels being higher than 4 mmol/L with a 0.90

likelihood. Consequently, the most likely blood glucose levels

may turn out to be higher than the ‘‘ideal’’ target level of

�4.5 mmol/L. This safety mechanism helps prevent hypogly-

caemic episodes, reinforcing patient safety.

In summary, the stochastic parameter model developed can

be used to create virtual patients and trial simulations,

providing a platform for exploring different control protocols

and algorithms. Its direct clinical use enables glycemic control

interventions to be selected with a guaranteed likelihood of

exceeding a specified minimum glycemic value or target. These

features are enabled by the stochastic model and its use to

obtain a distribution of glycemic outcomes for a given SI n and

intervention.

4. Results and discussion

The glucose–insulin system model presented in Eqs. (1)–(3),

together with the stochastic parameter model developed,

defines the probability distribution of blood glucose levels

1 h following an intervention. Its applications are examined in

numerical simulations, using both retrospective clinical trial

data, and stochastic model generated data that imitate typical

ICU patient behaviours.

4.1. Clinical trial review

The stochastic model was verified retrospectively against

eight previous clinical control intervention trials performed in

the Department of Intensive Care Medicine, Christchurch

Hospital [35]. Blood glucose probability intervals were

produced at each control intervention and compared against

the measured values 1 h later. The results are summarised in

Table 2.

Retrospective clinical data review revealed the probabilistic

information on the trials performed. The SI stochastic model

can account for 84% of measurements over time with a 0.90

confidence, and 45% with a 0.50 confidence, over the eight

Table 2

Retrospective probabilistic assessment on clinical control trials

Clinical control

patients

Number of

interventions

Meas

inter-

1 9 2 (2

2 9 5 (5

3 9 1 (1

4 9 1 (1

5 9 7 (7

6 9 8 (8

7 9 5 (5

8 23 10 (4

Total 86 39 (4

clinical control trials used for analysis in the retrospective

study.

The simulated result for Patient 2 is shown in Fig. 7. The top

panel displays blood glucose, where the crosses are the actual

clinical measurements with 7% measurement error, the solid

line is the fitted blood glucose profile, and the circles are the

most likely probabilistic blood glucose predictions following

control interventions. The 0.90 and inter-quartile probability

intervals are also shown with each most probable blood glucose

forecast. The bottom panel shows the fitted SI and the

probabilistic bounds produced from its stochastic model.

Patient 2 represents a typical insulin resistant critical care

patient, whose fitted SI tends to be in the lower physiological

population range. The fitted SI between 120 and 180 min

departed significantly from the predicted SI, reflecting a sudden

hyperadrenergic event that extensively altered the patient

condition, which was an episode of atrial fibrillation around

150 min. Consequently, the probabilistic prediction made for

180 min fails to agree with the actual measurement. The results

from this patient demonstrated the sufficiency of the stochastic

model. Most blood glucose levels were within the 0.90

confidence bands. The outlying events at 120 and 180 min were

due to more extreme variations, which are not uncommon in the

critically ill. However, to capture these events, the confidence

bands will become meaninglessly wide. Hence, the 0.90

confidence level was chosen for practicality and usefulness in

decision support.

Fig. 8 shows the same form of results for Patient 4. This

patient began the blood glucose regulation trial at a

normoglycemic level. This patient’s blood glucose briefly

dropped below 4 mmol/L during the clinical trial, which was

not initially accounted for with the probabilistic forecast.

Different control interventions were then explored in Fig. 9

using the SI stochastic model to assist decision making. A

comparison between the clinical trial (panels A and C) and the

new control intervention using the confidence intervals (panels

B and D) is shown in Fig. 9. These panels indicate that using the

distribution of SI enables more effective decision making and

control.

The simulated new control protocol aimed to maintain the

0.90 confidence intervals above 4 mmol/L. More aggressive

control interventions were thus taken in the first half of the trial,

resulting in the blood glucose levels more tightly maintained at

urement error within

quartile confidence interval

Measurement error within

0.90 confidence interval

2%) 7 (78%)

6%) 7 (78%)

1%) 7 (78%)

1%) 6 (67%)

8%) 9 (100%)

9%) 8 (89%)

6%) 9 (100%)

3%) 19 (83%)

5%) 72 (84%)

Fig. 7. Simulated clinical control trial on Patient 2.



Fig. 9. Clinical trial vs. simulated new control results on Patient 4.


a lower level up to 300 min. The brief blood glucose excursion

below 4 mmol/L was still unavoidable because the change in

the patient’s SI exceeded the 0.90 confidence band limit of the

created SI stochastic model. However, a more vigorous

remedy action was taken at 360 min in using the stochastic

model confidence intervals to result in the blood glucose levels

having a 0.90 confidence of being above 4 mmol/L in 1 h.

Overall, the application of the SI stochastic model in control

protocols, as in this case, delivers tighter and safer glycemic

management.

Fig. 10 shows the simulated results for Patient 8. This patient

was the first 24-h clinical control trial performed, with a

measurement interval of 1 h [35]. Out of 23 predictions, 7 were

outside of the inter-quartile range, but within the 0.90

probability interval. The fitted SI profile shows the evolution

of the patient condition through a day. The Markovian SI

stochastic model successfully predicts the SI variation trend,

shown by the shifting of the SI probability intervals closely

following the identified SI.

Note that when SI increases, the probability interval for the

resulting blood glucose levels also tightens. The wide range of

uncertainties in blood glucose levels associated with very low SI

values reflects a common problem in critical care, where highly

insulin resistant patients with high insulin inputs, as often seen

in intensive insulin therapy, can experience a sudden plunge in

blood glucose levels and become hypoglycaemic [22,23,25].

This type of situation was also encountered by Patient 4 in

Figs. 8 and 9, whose SI profile also was in the lower

physiological range [26,43].

4.2. Virtual control trial results

‘‘Virtual patients’’ with pG and SI following the stochastic

behaviour of the Markov model developed reflect typical

critical care patient conditions. A virtual cohort of 200 patients

was created and tested in simulated trials. Initial conditions of

these virtual patients, including starting blood glucose levels,

initial SI levels, insulin infusion, dextrose infusion, etc., were

randomly chosen to represent typical critical care situations.

Resulting blood glucose probability intervals from control

inputs are produced with each control intervention. The

simulated trials each span 24 h and 23 hourly blood glucose

measurements (excluding the starting blood glucose levels)

were analysed against the probability intervals. The results

from the simulated trials are summarised in Table 3.

The virtual cohort produced results that are similar to the

eight clinical trials. The inter-quartile confidence intervals

covered 46.04% of blood glucose measurements, and the 0.90

confidence intervals covered 87.85%. The defined hypogly-

caemic level for the trials was 4 mmol/L. All control

interventions maintained a minimum of 0.90 confidence level

in the resulting blood glucose levels being above 4 mmol/L.

Across the 200 virtual patient cohort (4600 measurements), 2

measurements (0.04%) fell below 3 mmol/L (2.6 and

2.9 mmol/L), and 111 (2.41%) fell below 4 mmol/L.

These slightly hypoglycaemic events all occurred when SI

took sudden rise that exceeded the 0.90 probability intervals in

SI. More specifically, they are similar to clinical results reported

for a very similar control trial over 1500 patient hours, which



reported 1.6% below 4 mmol/L with a minimum of 3.2 mmol/L

[48]. Hence, these results match ongoing clinical results with a

very similar cohort, providing some further validation.

The targeted, confidence interval-based control algorithm

was able to maintain blood glucose levels within the 4–6 mmol/

L normoglycemic bands 69.62% of the time once the blood

glucose levels were brought into this range. This value exceeds

clinical trials without confidence bands assisted targeted blood

glucose control [35,48]. Out of the cohort of 200 virtual

patients, 41 (20.50%) never achieved normoglycemia at any

instance during the 24 h trials due to insulin resistance and

insulin effect saturation. These patients have virtual SI profiles

generally in the very low physiological range. Consequently,

insulin-stimulated glucose removal was constantly saturated

with high insulin doses, and the blood glucose confidence bands

are also wide. Therefore, to maintain a 0.90 confidence level

against a hypoglycaemic event, the achievable blood glucose

Table 3

Virtual trial results

No. of hourly blood

glucose levels within

4–6 mmol/L (%a)

No. of hourly blood

glucose levels below

3 mmol/L (%b)

No. of hourl

glucose level

4 mmol/L (%

Maximumc 23 (4.76%) 1 (4.35%) 5 (21.74%)

Meanc 10.32 (69.62%) 0.01 (0.04%) 0.56 (2.41%)

Minimumc 0 (100.00%) 0 (0.00%) 0 (0.00%)

S.D.c 7.74 (25.59%) 0.10 (0.43%) 0.89 (3.87%)

a Percentage of time blood glucose levels stayed within 4–6 mmol/L once bloodb Total number of hourly blood glucose levels excluding the starting blood glucc Virtual cohort size n = 200.

reduction is limited and the resulting glucose levels are higher.

These results match clinical observations that blood glucose

levels under control are more volatile in highly insulin resistant

patients [13,26,35].

The average number of hours taken for the blood glucose

levels to be reduced to within 4–6 mmol/L is 6.27 h from the

start of the trial for those who ever entered this range. This

supports the need of long-term blood glucose control in

providing sufficient blood glucose management [13,35,36]. In

those patients whose blood glucose levels were ever reduced to

4–6 mmol/L (n = 159), 125 (78.62%) stayed in the band for

greater than 50% of the time, and 81 (50.94%) stayed in the

band for greater than 75% of the time.

In summary, the glucose–insulin system model, together

with the integral-based parameter identification, can effectively

capture critical care patient behaviour. In addition, the

stochastic model further enhances the ability to predict, as

y blood

s belowb)

No. of hourly blood glucose

levels within the inter-quartile

confidence intervals (%b)

No. of hourly blood glucose

levels within the 0.90

confidence intervals (%b)

19 (82.61%) 23 (100.00%)

10.59 (46.04%) 20.21 (87.85%)

4 (17.39%) 13 (56.52%)

2.44 (10.60%) 1.63 (7.09%)

glucose levels had reduced to �6 mmol/L.

ose level = 23.


well as imitate, typical critical care patient dynamics. The

incorporation of the stochastic model into the numerical

glucose–insulin system can also be used to create ‘‘virtual

patients’’, which present a platform to experiment with

different clinical control protocols that use probabilistic

confidence intervals based on clinically observed patient

dynamics and evolution. Higher identified SI levels result in

tighter blood glucose probability intervals, making tighter

control easier with less control effort. However, blood glucose

probability intervals widen at lower SI levels, limiting the

accuracy of tight glycemic regulation that can be obtained.

The percentages of blood glucose levels within the 0.90 and

0.50 confidence intervals are just slightly less than the specified

confidence. This may be due to the fact that the stochastic

model is built on first order SI variation only. Higher order

variation relationship may possibly exist which requires further

investigations. In addition, the variability in pG was ignored,

which might have compromised some fitting and predicting

accuracy. Nevertheless, the confidence intervals captured both

the clinical and the simulated results effectively.

5. Conclusions

The stochastic model defines the distribution of blood

glucose levels 1 h following a known glycemic control

intervention, and thus enables more knowledgeable and

accurate prediction for glycemic control. The model created

was evaluated on 8 prior clinical trials and 200 virtual patient

trials. The overall results agreed with the confidence intervals.

The stochastic model acts as a tool to assist clinical intervention

decisions, maximise the probability of achieving desired

glycemic regulation, while maintaining patient safety. The

impact of control inputs on probabilistic blood glucose results

can be assessed, giving confidence in the effectiveness of the

control protocol against evolving patient dynamics, particularly

with respect to avoiding hypoglycaemia.

The quality of blood glucose control is closely linked with

patient condition, in particular with respect to insulin

sensitivity. Higher identified SI levels give tighter blood

glucose probability intervals, making tighter and safer control

possible with subtle control efforts. Blood glucose probability

Fig. A1. Sample space orthonormalisation. Panel A is the o

intervals widen at lower SI levels, limiting the accuracy of tight

glycemic regulations. Caution against sudden reduction in

glycemic levels is needed at typically low levels of SI, where

significant doses of insulin are administered, while the range of

possible change in blood glucose levels is broad. In addition,

‘‘virtual patients’’ created from the stochastic model presents a

platform to experiment different clinical control protocols with

a probabilistic knowledge based on clinically observed

evolving dynamics. Simulated control inputs can thus be

evaluated on realistic virtual patient dynamics driven by the SI

profiles.

Appendix A

In this research, the form of the kernel chosen for the kernel

density estimation method is the Gaussian kernel, f, as defined

in Eq. (7). The variance s of the kernel depends on the local data

density such that the shape of the kernel is optimised to produce

smooth approximation of the true data behaviour. To define the

local data density, standard orthonormalisation is performed.

Let x = SI n and y = SI n+1, n = 1, 2, . . ., k, totalling a data set of k

samples. Both x and y are column vectors. The orthonormalisa-

tion steps are as follows:

(1) S

rigin

olve for the covalence of ½ x y �:

C ¼ covð½ x y �Þ (A1)

(2) P
erform Cholesky factorisation on C:
C ¼ RRT (A2)

(3) C
alculate the transformation matrix A:
A ¼ ðRTÞ�1(A3)

(4) C
entre the samples and perform space transformation with
A:

x ¼ x� x (A4)

y ¼ y� y (A5)

½ x y � ¼ A � ½ x y � (A6)

The samples before and after orthonormalisation are shown

in Fig. (A1).

al space and panel B is the orthonormalised space.


(5) D
efine the spread of the samples in the transformed space
by calculating the maximum distance between the furthest

sample and the origin. (Also see Fig. (A1), panel B).

R ¼ maxðffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

qÞ (A7)

(6) C
alculate the variance sx and sy:
Sx ¼ min

�stdðxÞ; IQRðxÞ

1:348

�(A8)

Sy ¼ min

�stdðyÞ; IQRðyÞ

1:348

�(A9)

sz ¼ SxðmR2k1=3Þ�1=6(A10)

sy ¼ SyðmR2k1=3Þ�1=6(A11)

where m is the no. of samples within a radius of k�1/6 from

corresponding entries of ½ x y �.

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Documents

Stochastic modelling of insulin sensitivity variability in critical care