14th IFAC (International Federation of
Automatic Control) Symposium on System
Identification, SYSID 2006, March 29-31
IMPACT OF SYSTEM IDENTIFICATION METHODS
IN METABOLIC MODELLING AND CONTROL
Dr. J. Geoffrey Chase
Department of Mechanical Engineering
Centre for Bio-Engineering
University of Canterbury
Christchurch, New Zealand
The Situation

Metabolic modelling can significantly improve the clinical control of
hyperglycaemia with model-based protocols (e.g. Hovorka et al., 2004;
Chase et al., 2005)

For clinical utility, model parameters must be accurately identified
for real-time prediction of response to intervention

Current identification methods are mostly non-linear and nonconvex, and very computationally intense


With increasing model complexity, parameter trade-off can result
in problematic identification. A typical solution is probabilistic
population fitting methods (e.g. Vicini and Cobelli, 2001; Hovorka et al., 2004)
Typical clinical situation might use models and identification
methods from different sources with local cohort/data.
The Problem & The Goal

Non-linear and non-convex identification methods and models can
deliver sub-optimal results, affecting control prediction
– Clinically, prediction is the only true measure of utility

What is the clinical impact of mixing models and identification
methods (if any)?
– Currently, model, system ID method and control are all designed together.
– What happens if someone “mix and matches” without the original designers
insights or experience?

This research compares a recently introduced linear, convex integralbased method and the commonly used non-linear recursive least
squares (NRLS) identification method
– Using an accepted metabolic system model from one source and clinical data
from another source for “independence”
– “Independence” represents the typical clinical situation and avoids the
models or methods being tuned for the cohort

The goal is to examine the computational cost and outcomes of these
different methods in a clinical control application context
Model

The model chosen for comparison is loosely based on the 2compt. minimal model (2CMM) first proposed by Caumo &
Cobelli (1993)
– Well documented model that is widely used as a foundation

Main change is the 3 insulin compartments for the remote
effects of insulin on glucose distribution/transport, disposal
and EGP introduced by Hovorka et al. (2002)
– Similar model has been used clinically for control

Comprises 6 compartments in total
– 2 glucose compartments g1(t) and g2(t)
– 3 insulin action compartments QD(t), QT(t) and QEGP(t)
– 1 plasma insulin compartment I(t) )
Integral-Based Parameter Fitting




A “minimal” approach to identification is used with most model constants
identified a priori from literature results
–
Selection of population valued constants is a major issue in biomedical modeling as it
assumes the parameter is not highly sensitive to results
–
This assumption may not be true in all clinical scenarios or cohorts
–
Required in many cases to ensure the model is identifiable from the available data
The remaining insulin sensitivities SI,D, SI,T and SI,EGP are identified as time-varying
model parameters driving the model dynamics (details in the paper)
This approach minimises total computational cost while enabling individual model
constants to be varied for more optimised prediction and fit (e.g. Hann et al., 2005)
What is the effect of mixing this approach and this model?
–
Would be an “easy” combination for an independent researcher
–
Will all assumptions on constant parameters hold?
–
Can we identify despite inaccessible, unmeasurable compartments?
Integral-Based Parameter Fitting

SI,D, SI,T and SI,EGP are defined piecewise constant over a time period
of 60mins using Heaviside step functions, H(t).
N
S I , j   S I , j ,i ( H (t  t( i 1) )  H (t  ti )) where j  D, T and EGP
i 1

Definition of the distribution of these parameters are arbitrary i.e. cubic,
quadratic etc.
– Approach allows constants to define variation and be pulled out of integrals

2nd order polynomial interpolation is assumed between glucose
measurements in the accessible glucose compartment g1(t)
– Error using this approximation has been shown to be minimal
(Hann et al., 2005)
Integral-Based Parameter Fitting



Inaccessible glucose compartment g2(t) modelled using a 2nd order
Lagrange polynomial approximation to analytical solution for this
immeasurable compartment (fortunately, it’s a simple enough dynamic)
Within a time period of [t0 tf ], an arbitrary number of equations can be
generated by integration of model equations over different time periods
The non-linear model thus decomposes into a linear equation system in
unknown constants defining parameters to be identified
– Resulting least squares solution is starting point independent and convex!
A S I ,T g2 (t0 ) g2 (t1 ) g2 (t f ) S I , EGP EGPb   b
T
C S I ,D  d
Clinical Data

Patient data (n=7) was chosen from an intensive care unit
hyperglycaemia control trial

(Chase et al., 2005)
Each set of patient data spans 10hrs with glucose measurements at
0.5hr intervals.
– Average glucose levels are ~ 6mmol/L (range ~4-10 mmol/)

Prediction window is 1hr following hourly clinical interventions

Median APACHE II = 23, inter-quartile range = 19-25
Results: Model Fit
Residual plot of model fit to patient data
2

1.5
Model fit errors
– Patient 2 (highest RMSE
1
0.80mmol/l, error SD
0.5
0.59mmo/l)
0
– Patient 5 (smallest RMSE
-0.5
0.15mmol/l, error SD
-1
0.08mmol/l)
-1.5
Patient 2
Patient 5
-2
-2.5
0

Patient 1,3,4,6,7
100
200
300
400
500
600
Model fit mean absolute percent error (MAPE) for cohort
ranges from 2.4-7.4% which is within reported sensor error
Results: Prediction
Residual plot of model prediction to patient data
4

3
– MAE for cohort is
2
1.03mmol/l, error SD is
1
0.78mmol/l
0
– RMSE is 1.31mmol/l, MAPE
-1
20.21%
-2


– Very variable depending on
Patient 2
Patient 5
Patients 1,3,4,6,7
-3
-4
100
Model prediction errors
150
200
250
300
350
400
450
500
550
the patient and/or time
600
Prediction MAPE exceeds the reported sensor error
Errors are mostly at or within sensor error or very wide
Results: NRLS
Average model fit RMSE for NRLS and integral-based methods
0.7

Nonlinear Parameter ID
Integral-Based Parameter ID
0.6
NRLS implemented using a non-linear
ODE least squares solver in MATLAB on
a Pentium M 1.7GHz PC, 1Gb RAM
0.5

0.4
Integral method has lower error even
with approximated compartment
0.3

0.2
0.1

0
0
100
200
300
400
500
600
Average values of SI,D, SI,T and SI,EGP
from literature used as starting points
Integral-based method with linear
approximation of g2(t) is 140X-660X
faster than NRLS

NRLS finds local minima as seen in higher average model fit RMSE at most times

Average time to complete model fit for one 10hr trial using linear integral-based
method was 0.46±0.16s vs 122.60±42.81s using NRLS
Is it the model or method?
Average model prediction RMSE with 1-compt. glucose model
1.8
1.6

2-compartment glucose model
Chase et al. model
(1-compartment glucose model)

1.4
1.2
1

0.8
(Chase et al., 2005)
Care must be taken not to over fit
available data with model dynamics.
For this cohort, the 1-compt. glucose
model has significantly smaller prediction
errors for a given set of parameters
This result is due to differences in model
dynamics and ability to fit the observed
behaviour, independent of fitting method
0.6

However, model constants were average a
priori values and not further optimised

Hence the level of prediction accuracy
reported may be expected
0.4
0.2
100

150
200
250
300
350
400
450
500
550
600
A convex identification method exposes the model prediction errors,
identifying potential inadequacies in model dynamics and/or constants
Some Conclusions


Cohort model fit RMSE and MAPE were lower using linear integral-based method
compared to NRLS – for the same model
Model complexity can be extended (i.e. multiple compartments) without
significantly affecting identification computation time Integrals can be used for
simple inaccessible compartments using approximations

Fitted parameters were all within reported physiological ranges

Issues:
–
–
Different model dynamics and parameters may work better for different cohorts or
situations – the comparison is not “complete” and this work is presented to show the
potential impacts
A priori global identifiability should always be considered. Not all models are globally
identifiable for all parameters.

Linear, integral-based method shown to have lower computational cost leading to
increased PI speed

A convex method can identify potential areas of model difficulty or which other
parameters may need to be identified in place of a population value.
Acknowledgements
Jessica Lin & AIC3
AIC1
Jason Wong & AIC4
Thomas Lotz
The Danes
Prof Steen
Andreassen
AIC2 & Dr. Geoff Shaw
Dunedin
Dr Kirsten
McAuley
Maths and Stats Gurus
Dr Dom Lee
Dr Bob
Broughton
Prof
Graeme Wake Dr Chris Hann
Prof Jim Mann