DYNAMIC AUTOREGRESSIVE MODELLING OF CRITICAL

CARE PATIENTS AS A BASIS FOR HEALTH MONITORING

K. Van Loon

, G. Meyfroidt

, T. Tambuyzer

, G. Van den Berghe

, D. Berckmans

and J.-M. Aerts

Division Measure, Model & Manage Bioresponses, Katholieke Universiteit Leuven

Kasteelpark Arenberg 30, B-3001, Leuven, Belgium

Department of Intensive Care Medicine, University Hospital Gasthuisberg, Herestraat 49, B-3000, Leuven, Belgium

Keywords: Critical Care Patients, Health Monitoring, Time Series Analysis, Autorgressive Modeling.

Abstract: Real-time modelling techniques could be valuable to continuously evaluate individual critically ill patients

and to help the medical staff with estimation of prognosis. This preliminary study examines the possibilities

to distinguish survivors from non-survivors on the basis of instabilities in the dynamics of daily measured

variables. A data set, containing 140 patients, was generated in the intensive care unit (ICU) of the

university hospital of Leuven. First and second order dynamic auto-regression (DAR) models were used to

quantify the stability of time series of three physiological variables as a criterion to distinguish survivors

from non-survivors. The best results were found for blood urea concentration with true negative fractions of

45/72 (63%) and true positive fractions of 43/68 (62%). The results indicate that the dynamics of time series

of laboratory parameters from critically ill patients are indicative for their clinical condition and outcome.

1 INTRODUCTION

Physicians have for long recognized the importance

of considering the temporal dimension of illness for

arriving at a diagnosis and deriving treatment

strategies (Belair et al., 1995). The study of disease

dynamics, or how disease states change with respect

to time, is providing a key to understanding

abnormalities in underlying physiologic control

mechanisms (Goldstein et al., 2003). For monitoring

purposes, especially changes in dynamic

characteristics seem to be relevant in distinguishing

health from disease (Glass, 2001; Buchman, 2004;

Van Loon et al., 2010). This indicates the potential

of approaches that aim at quantifying dynamic

characteristics of individual patients on-line during

their stay in the intensive care unit (ICU). Several

attempts to take the time-varying aspect of the health

status of critically ill patients into account have been

reported (e.g. Chang et al., 1988; Clermont et al.,

2004; Toma et al., 2007, 2008).

In most of these studies, the available

information at a certain instance in time was

summarised in one score and the calculation of this

score was repeated in time. Afterwards, a

classification or prediction model was built using

these summary variables. Instead of using repeated

scores, it is also possible to extract dynamically

relevant features from the commonly measured

physiological data itself. A large number of

variables are continuously monitored and stored in

the ICU environment.

A candidate approach for monitoring individual

patients in the ICU is time series analysis. A few

investigations have employed the use of time series

analysis in the field of intensive care medicine (e.g.

Lambert et al., 1995). It has been shown that time

series analysis techniques are suitable for

retrospective analysis of physiological variables. A

computationally similar, but more challenging task

is the on-line analysis of intensive care monitoring

data (Imhoff et al., 1999).

The objective of the reported research was to

explore whether recursive time series analysis can

be used to monitor individual patients in the ICU.

More specifically, the aim was to test if the

occurrence of temporal instabilities in the dynamics

of time series of continuously measured

physiological data contains valuable information for

distinguishing between survivors and non-survivors.

Van Loon K., Meyfroidt G., Tambuyzer T., Van den Berghe G., Berckmans D. and Aerts J..

DYNAMIC AUTOREGRESSIVE MODELLING OF CRITICAL CARE PATIENTS AS A BASIS FOR HEALTH MONITORING.

DOI: 10.5220/0003784800850090

In Proceedings of the International Conference on Bio-inspired Systems and Signal Processing (BIOSIGNALS-2012), pages 85-90

ISBN: 978-989-8425-89-8

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

2 MATERIALS AND METHODS

2.1 Patient Database

The database was derived from a larger database

(1200 patients) created for a clinical study with a

similar setup and purpose in a group of critically ill

patients from a medical ICU (Van den Berghe et al.,

2006). In order to have enough data points for time

series analysis we had to select patients with a

length of stay of at least 20 days because in the

existing data sets, most of the stored data were

available only once daily. A second selection

criterion was that we only accepted patient data with

no missing measurement values. A set of 140

patients fulfilled both criteria.

An overview of the used data set is depicted in

Table 1. The protocol of this trial was approved by

the ethical commission of our hospital.

Table 1: Characteristics of the database.

Number of patients 140

Age (mean ± std) 62 ± 14

Sex M: 95, F: 46

BMI 25 ± 5

We used daily measurements of three variables,

namely: maximum body temperature (Tmax, °C),

white blood cell count (WBC, 10

/L) and urea

concentration (Uconc, mg/dl).

2.2 Recursive Modelling

Since the physiological patient responses are time

variant, most of the measured physiological

variables are non-stationary, in the sense that the

statistical properties of the signal are changing

slowly over time in relation to the rates of change of

the stochastic state variables in the system under

study. When the system is non-stationary, models

with time varying parameters should be used for the

analysis (Pedregal et al., 2007). In this study we

used a Dynamic Auto-Regression (DAR) model

which can be formulated as:







(



,)





(1)

in which 

(





,

)

=1+







+







+⋯+









is a time variable parameter polynomial in

the backward shift operator 



; y

is the considered

physiological variable; e

is zero mean white noise.

Here, the adjective ‘dynamic’ means the model

has time variable parameters and not that the DAR

model is dynamic in a systems sense. A random

walk model was specified and a time domain

maximum likelihood estimation was applied to find

the optimal parameter estimates.

By multiplying equation (1) throughout by



(





,

)

the DAR model in the time series

formulation is obtained:





=−







−⋯−







+



(2)

From this equation it can be seen that 



calculated from previous samples of itself plus a

random component in the form of the white noise 



For more details, reference can be made to Taylor et

al. (2006) and Pedregal et al. (2007).

In a first step, first and second order DAR

models were computed for the three variables of the

development data set, since in preliminary analysis

these model orders led to the best results in terms of

the Akaike’s Information criterion (AIC). The first

and second order model structures were as follows:





=−







+



(3a)





=−







−







+



(3b)

On the basis of the correlation coefficients between

the measured variables and the one-step-ahead

predictions, the best performing variables were

selected for further analysis in the second step.

Average correlation coefficients < 0.70 were

considered as not sufficient in order to have an

accurate model.

A further selection in the variables and model

orders was made considering the uncertainty of the

parameters in the calculated models. The average

standard error (SE) on the parameters was calculated

and used as a measure for the reliability of the

model. In the recursive algorithm, for all

consecutive estimations of the model parameters, the

relative standard error (RSE) of the parameters was

calculated using following equation:

 =





 

 × 100(%)

(4)

The average of the relative standard errors was taken

from day 15 until the end of the data set for each

patient. The recursive algorithm needed about 14

data points to result in a reliable model, so the 14

first values were not considered.

2.3 Model-based Classification of

Survivors vs. Non-survivors

The hypothesis of this work was that a patient that

becomes unstable at least once during his/her stay in

the ICU, will not survive. When measured variables

BIOSIGNALS 2012 - International Conference on Bio-inspired Systems and Signal Processing

become unstable, this does not imply an increase in

the variability or the irregularity of the signal. A

higher variability often even corresponds to more

healthy conditions (Lipsitz, 2002).

To have a stable system, in this case the patient,

it is required that all poles of the transfer function lie

inside the unit circle. For a first order system, this

means that the following criterion should be met for

the a

-parameter in equation (3a) (Box et al., 1994):

−1 < 



(5)

The time course of the measured urea concentration

values (as an example) and the calculated stability

criterion (equation 5) for two example patients are

shown in Figure 1 for a survivor and a non-survivor.

For a second order model, the criteria for

stability can be expressed in terms of the two model

parameters in the following way (Box et al., 1994):





+



>−1





−



>−1

(6)

−1 < 



where 



and 



are the model parameters as

described in equation (3b). When one of these

criteria is not met, the system is unstable.

On the basis of the stability criteria, the

classification between survivors and non-survivors

was made and quantified in terms of true positive

fractions (sensitivity) and true negative fractions

(specificity). The true positive fraction (TP) was

defined as the fraction of patients that becomes

unstable at least once during the stay in the ICU and

died. The true negative fraction (TN) was defined as

the fraction of patients that did not become unstable

and survived. Because the recursive parameter

algorithm needed about 14 data points (days) of past

data to produce reliable parameter estimates, the

stability test was performed from day 15 on. This is

a drawback of the used methodology, but the aim of

this preliminary study was to investigate whether the

stability of measured physiological variables of the

patients gives valuable information about the

patients’ survival when they have a long ICU stay

and not to predict the outcome as soon as possible

after arrival in the ICU.

3 RESULTS AND DISCUSSION

In the first step, first and second order models were

calculated for the three variables and the correlation

coefficients between the measured variables and the

one-step-ahead predictions were analysed. Table 2

gives an overview of the average correlation

coefficients for each variable for the first as well as

for the second order models. The results indicate

that the variable Tmax could not be modelled

accurately (average correlation coefficients < 0.70).

Consequently, only the variables Uconc and WBC

were selected for further analysis.

Table 2: The correlation coefficients between the

measured variables and the one-step-ahead predictions of

the first and second order dar models calculated on the

variables tmax, uconc and wbc.

Tmax Uconc WBC

order 0.51 0.86 0.73

order 0.54 0.85 0.77

Secondly, the reliability of the parameters was

examined in terms of their relative standard errors.

The average errors are given in Table 3 for the first

and second order models of the two remaining

variables. From this table it can be seen that the

errors on the parameters of the second order models

were always larger than those of the first order

models. Therefore it was decided to disregard the

second order models from this step on.

Table 3: The mean relative standard errors (MRSE) on the

parameters of the first and second order DAR models

calculated on the variables Uconc and WBC of the

development data set (%).

Model Parameters

MRSE

Uconc

(%)

MRSE

WBC

(%)

order a

43.15 39.16

order a

486.24 386.72

1036.58 909.44

The recursively calculated parameter estimates were

tested against the stability criteria (equation (5)) in a

third step. Table 4 summarizes the calculated true

positive fractions and the true negative fractions for

the first order models of the two remaining

variables.

Table 4: The TN and TP values for Uconc and WBC.

Uconc WBC

TN 45/72 (63%) 61/72 (85%)

TP 43/68 (62%) 22/68 (32%)

If we were to base clinical decisions upon a model,

it would be clinically more acceptable to classify a

non-survivor erroneously as a survivor than to

classify a survivor as a non-survivor. So, preferably

the TN is at least as big as the TP and both the TP

DYNAMIC AUTOREGRESSIVE MODELLING OF CRITICAL CARE PATIENTS AS A BASIS FOR HEALTH

MONITORING

and TN should be as close to 100% as possible.

Considering this, the best result was obtained for

Uconc with a TN of 45/72 (63%) and a TP of 43/68

(62%). This signal was consequently considered as

the most indicative variable for outcome prediction,

although these results are not good enough to be

useful in clinical practice.

Uconc turned out to be the best choice for the

classification of survivors versus non-survivors. The

TN for WBC was better than in the case of Uconc,

but the TP was a lot worse. When looking at the

reliability of the parameters, the relative standard

errors on the parameters of the models WBC were

lower than the errors for Uconc, but the former have

a very low TP. Consequently, Uconc is the best

option for the given study purpose.

All selected variables are used in the clinical

setting as markers of inflammation or organ function

and therefore it is not surprising that they are

predictive for mortality in this subset of patients

with a prolonged stay in the ICU. The blood urea

concentration was found to be a prognostic marker

in several types of patients (Beier et al., 2011). In

the study of Jackson et al. (2008) it is shown that an

elevated urea concentration is more powerful than

the estimated glomerular filtration rate at predicting

an increased risk of early mortality following

admission with heart failure. In this study a Cox-

proportional hazard model of log[urea] (per unit

change) resulted in a hazard ratio for risk of death of

1.79 (± 95% CI 1.08-2.97, P = 0.003). An increased

postoperative serum urea concentration is also

associated with an increase in 30-day mortality in

patients undergoing emergency abdominal surgery

(odds ratio 4.79, ± 95% CI 2.37-9.70, P = 0.003;

Harten et al., 2006). The relevance of blood urea

nitrogen (BUN) as a marker for length of stay and

mortality at the intensive care unit for patients with

acute necrotizing pancreatitis was investigated by

Faisst et al. (2010). In their study, these authors used

thresholds on absolute values of BUN as a predictive

value. When using a threshold (cutoff) value of 33

mg/dl, high BUN levels correctly predicted a

prolonged length of stay in 89% of the cases

(positive predictive value, PPV) and the negative

predictive values (NPV) for BUN on admission and

in the course of the disease was 62% and 77%. With

the same threshold, mortality could be correctly

ruled out in 82% of the cases on admission and in

92% of the cases in the course of the disease (NPV).

The PPVs were lower with 67% on admission and

56% in the course of the disease. In addition to these

studies, our results indicated that not only the

absolute steady state levels of Uconc, but also its

dynamics, can be predictive for survival in the ICU.

To the authors’ knowledge no studies have been

performed using time series dynamics of urea

concentration in critically ill patients at the ICU in

relation with mortality.

(a)

(b)

(c)

(d)

Figure 1: Time courses of the daily measured urea

concentration values for a survivor (a) and non-survivor

-parameter

values for a survivor (b) and non-survivor (d). The vertical

dashed line in (b) and (d) indicate the end of the period of

the first 14 data points (days) that are needed for reliable

parameter estimation. The horizontal line indicates the

threshold of a

= -1.

BIOSIGNALS 2012 - International Conference on Bio-inspired Systems and Signal Processing

The occurrence of instabilities in the measured

biological signals of (most) non-survivors might be

explained by the fact that, in critical care patients

before dying, the complex closed-loop responses

that operate to keep the organism in equilibrium may

not be evolved to produce a healthy response to

stress, and thus respond in a maladaptive way. This

fits with the work of Lipsitz (2002) who connects

the proper working of physiological systems with

the stability of its dynamic response. However, it

can be expected at one hand that some patients

become dynamically unstable but recover from this

situation or on the other hand that some patients can

die due to very acute problems (e.g. brain

haemorrhage) which are not preceded by periods of

instability in the measured dynamics. Therefore, the

findings in our study need to be validated and

confirmed in a larger sample of patients to evaluate

the concept thoroughly.

In ICU's worldwide, attempts to improve data

processing have centered on computerized systems

and several patient data management (PDMS)

systems have been developed (Toma et al., 2007).

This is software where virtually all patient and

therapy related information is stored on a high

resolution basis. Consequently, there is a great need

for integrating the data and automating the

recognition of several diagnoses, since the quality of

health care systems depends on making the right

decisions at the right time and place (Fonseca et al.,

2009). Without automated systems, clinicians have

to manually extract the necessary information,

which is a time-consuming work that distracts them

from critical tasks and increases the risk of making

mistakes (Spencer et al., 1997). In combination with

recursive modelling techniques, such systems might

allow to model the patients’ dynamic responses in

real-time as a basis for improving personal health

status monitoring.

4 CONCLUSIONS

We found that the patients’ dynamics contains

interesting information when distinguishing between

survivors and non-survivors. A data set of 140

patients was used for the analyses. On the basis of

stability measurements calculated from the

parameters of recursive time series models on

physiological data, we were able to separate

survivors form non-survivors. The best results were

obtained when using blood urea concentration which

gave a true negative fraction of 45/72 (63%). The

true positive fraction was 43/68 (62%). The results

of this study need to be validated and afterwards

confirmed in larger trials, before the described

methodology could be considered in the future in

combination with patient data management systems

to support the physician in on-line monitoring and

decision taking for individual patients.

ACKNOWLEDGEMENTS

We wish to thank the Katholieke Universiteit

Leuven for funding the research reported in this

paper (Interdisciplinary Research project

IDO/03/006).

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BIOSIGNALS 2012 - International Conference on Bio-inspired Systems and Signal Processing