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Published in IET Systems Biology
Received on 17th January 2008
Revised on 29th May 2008
doi:10.1049/iet-syb:20080094
ISSN 1751-8849
Optimal experimental design with the
sigma point method
R. Schenkendorf A. Kremling M. Mangold
Max-Planck-Institute for Dynamics of Complex Technical Systems, Sandtorstr. 1, 39106 Magdeburg, Germany
E-mail: rschenke@mpi-magdeburg.mpg.de
Abstract: Using mathematical models for a quantitative description of dynamical systems requires the
identification of uncertain parameters by minimising the difference between simulation and measurement.
Owing to the measurement noise also, the estimated parameters possess an uncertainty expressed by their
variances. To obtain highly predictive models, very precise parameters are needed. The optimal experimental
design (OED) as a numerical optimisation method is used to reduce the parameter uncertainty by minimising
the parameter variances iteratively. A frequently applied method to define a cost function for OED is based
on the inverse of the Fisher information matrix. The application of this traditional method has at least two
shortcomings for models that are nonlinear in their parameters: (i) it gives only a lower bound of the
parameter variances and (ii) the bias of the estimator is neglected. Here, the authors show that by applying
the sigma point (SP) method a better approximation of characteristic values of the parameter statistics can be
obtained, which has a direct benefit on OED. An additional advantage of the SP method is that it can also be
used to investigate the influence of the parameter uncertainties on the simulation results. The SP method is
demonstrated for the example of a widely used biological model.
1 Introduction
The use of mathematical models to analyse complex systems is
very common in many research fields. One important step of
model development is parameter identification (PI), that is,
the determinationof unknown model parameters
minimising the difference between simulation results and
measurements. Even for the deterministic case with ideal
measurement, it is hard to obtain unique estimations of
parameters such as the maximum growth rate mmand the
substrate affinity constant KSof Michaelis–Menten kinetics
[1], which is frequently used for biological models. In practice,
the measurement is disturbed with noise leading to some
uncertainty of the estimated parameters [2]. This fact has a
strong impact on the model quality as only parameters with
small variances ensure simulation results with a highly
predictive power, which is necessary for process observation
and control or to study biological systems. The design of
appropriate experiments for PI is known as optimal
experimental design (OED) and is an important problem in
the field of systems biology [3, 4]. General overviews on OED
methodsfordifferentfieldsofapplicationcanbefoundin[5,6].
by
The OED mainly consists of two steps: (i) determining
the parameter variances from PI with available experimental
data and (ii) minimising these variances by stimulating the
system ‘optimally’ in a new experiment. These steps can be
reiterated until a sufficient accuracy is achieved (Fig. 1).
A frequently used method to obtain some information
about the statistics of the estimated parameters is based on
the inverse of the Fisher information matrix (FIM) [7].
If models are nonlinear with respect to their parameters,
the FIM may lead to a poor approximation of the
variances. To overcome this problem, various methods have
been developed to improve the calculation of parameter
uncertainties. In the majority of cases, the approaches are
based on Monte Carlo methods as the global sensitivity
analysis [3, 8, 9] and the Bootstrap approach [4, 10]. This
kindof realisationhasan
complexity, tending to prohibit their use in an iterative
OED process, as the variances of the parameters have to be
determined a lot of times. Here, we show that by applying
the SP method a better approximation of the parameter
statistics with a workable computational effort can be
obtained.
increased computational
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The basics of PI and OED are given in the next section.
Furthermore, the two more traditional methods (FIM and
Bootstrap) and the SP method for the determination of
parameter uncertainties are explained and their use in
the field of OED is assessed. After a successful demonstration
of the potential of the SP method by a simple bio-reactor
model, this model is extended by some intracellular
components to show that this novel approach is still
applicable for more complex models.
2
2.1 PI and model quality
Methods
Ordinary differential equations (o.d.e.’s) are widely used to
describe the dynamics of many systems. The o.d.e. system
dx(t)
dt
¼ f x(t), u(t), w(t), u, t
½?
(1)
and its solution
x(t) ¼ x(t0) þ
ðt
t0
f x(t), u(t), w(t), u, t½?dt
(2)
are functions of the dynamic states x(t) [ Rn, the system
input u(t) [ Rr, some process noise w(t) [ Rn, parameters
u [ Rpand time t.
The correlation between the states, inputs and the
simulation results ^ y(t) [ Rmis given by the output equation
^ y(t) ¼ h x(t), u(t), v(t), u, t
½?
(3)
where v [ Rmdenotes an additional measurement noise.
If the structure of the model, for example, given by the
stoichiometric network and the reaction kinetics, is known,
then the parameter vector u determines the predictive
power of the model. This can be expressed by the weighted
difference between the simulation results ^ y(t) [ Rmand
measurements y(t)
JML(^u) ¼1
2
y ? ^ y(x0, u,^u)
??TC?1
y
y ? ^ y(x0, u,^u)
??
(4)
The unknown parameters^u are estimated by minimising the
cost function equation (4) for a pre-defined input vector u(t).
In this work,^u denotes the estimated values of the unknown
model parameters, whereas u are the actual parameter values.
If the covariance matrix of the measurements is used for Cy,
then a maximum likelihood estimator results [2]. In practice,
the measurement noise v(t) and the process noise w(t)
lead to parameter uncertainties, expressed by the covariance
matrix
Cu¼ E
^u ? u
??
^u ? u
??T
??
(5)
where E ? ½ ? is the expected value of a random variable. In
addition, the nonlinearities in the model may lead to a bias in
the estimated parameters, that is, the expectation Mu¼ E[^u]
may be different from the actual parameter vector u. As the
‘true’ parameter vector u is unknown, an approximation of Cu
is needed to assess the estimation and if necessary to plan new
experiments with an increased information content. The mean
value Muand the matrix Cugive a measure for the accuracy or
quality of the model. If the covariances of the identified
parameters turn out to be unacceptably high, then one can try
to reduce them by choosing other experimental conditions,
that is, by making use of the additional degrees of freedom
given by the input vector u(t). This is the key idea of OED.
2.2 Reduction of the parameter
uncertainty by OED
The OED as a numerical optimisation problem reduces the
parameter uncertainty by the variation of some design
variables u(t). In order to formulate the optimisation
problem, it is necessary to define a scalar cost function F,
which depends on the parameters’ bias Mu? u and on
the covariance matrix Cu. In general, the bias is neglected
in the cost function, that is, F ¼ F(Cu). Well-known
optimality criteria [2] have been proposed for F
A-optimal design
D-optimal design
FA(Cu) ¼ trace(Cu)
FD(Cu) ¼ det(Cu)
FM(Cu) ¼ max
(6)
(7)
M-optimal design
ffiffiffiffiffiffiffiffiffi
Cu,ii
q
1
??
(8)
E-optimal design
FE(Cu) ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
lmax(Cu)
s
(9)
E?-optimal design
FE?(Cu) ¼lmax(Cu)
lmin(Cu)
(10)
with lmax(lmin) as the maximum (minimum) eigenvalue
of Cu. Which of them leads to the best result, depends on
Figure 1 Iterative nature of the OED process for a
two-dimensional parameter space
The information about the parameter variances, expressed by
confidence regions, is used to design new experiments that
minimise the parameter uncertainties, whereby the arrows
indicate the interaction between the experiments (measurements)
and the PI
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the model and the design parameters, hence no general
advice can be given. A challenging problem of OED is
how to obtain an accurate estimation of the covariance
matrix Cu
and of the estimated mean Mu. Three
approaches will be discussed in the following. Although the
first two are well-known, the third one has not yet been
used to identify parameter uncertainties directly.
2.3 Fisher information matrix
For the case of additive white measurement noise, the FIM is
determined by the following equation
FIM ¼
X
tk
ST
tk? C?1
y ? Stk
(11)
with the parameter sensitivity matrix S
Stk¼
@y1
@u1
????
tk
@y1
@u2
????
tk
???
@y1
@up
?????tk
@y2
@u1
????
????
tk
@y2
@u2
????
tk
???
.
...
........ ...
@ym
@u1
tk
???
@ym
@up?1
?????tk
@ym
@up
?????tk
2
6
6
6
4
6
6
6
6
6
6
6
6
6
6
3
7
7
7
5
7
7
7
7
7
7
7
7
7
7
(12)
the covariance matrix of the measurement data Cyand the
measurement time point tk. As the dynamic states x(t) are
time dependent, the following matrix differential equation
for the sensitivities has to be solved in parallel with the
dynamic system given in (1)
_S ¼@f
@y? S þ@f
@u;
S(0) ¼ 0p?p
(13)
The inverse of the FIM provides an estimation of the
parameter variances based on the Cramer–Rao ´ inequality [7]
Cu?
@E^u
h i
@u
FIM?1@E^u
h iT
@u
(14)
The equality only holds, if (i) the measurement errors are
additive and (ii) the model is linear in its parameters.
Furthermore, the FIM does not give any information on
E[^u]. Therefore in many cases, it is assumed in addition
that (iii) the estimate is unbiased, that is, E[^u] ¼ u. This
simplifies (14) to
Cu? FIM?1
(15)
Theuseof(15)isstateoftheartinthefieldofOED[2,11–14].
However, esp. conditions (ii) and (iii) are not met when
analysing complex biological models. Consequently, the
approximation FIM?1may strongly underestimate the actual
covariances [4]. To make things worse, FIM?1is not even a
reliable lower bound for Cu. If the gradient @E[^u]=@u
is small, the actual covariances can be overestimated by (15).
With an increasing measurement uncertainty, these effects
become more and more important depending on the
nonlinearity. In conclusion, the use of the FIM is
unsatisfactory for the OED of nonlinear biological models.
Alternative approaches are required.
2.4 Bootstrap
The roots of Bootstrap go back to the Monte Carlo method
[10]. A set of B fictitious measurement vectors ys
randomly. The set has to represent the statistics of the
measurement in the sense that the mean and the covariance
of the samples agree with the actual distribution
iis created
y ¼1
B?
X
1
B
i¼0
ys
i
(16)
Cy¼
B ? 1?
X
B
i¼0
(ys
i? y)(ys
i? y)T
(17)
A parameterestimation isdoneforeachofthesamplesys
leadstoasetofBestimatedparameters^ui.Tomakeastatement
about the confidence regions of the estimated parameters, the
percentile method is mostly preferred [4, 10], as it also
enables a determination of higher moments, for example,
the skewness of the random variables. However, the first two
moments, the mean and the variances, are sufficient for the
OED process and can be easily approximated by the
parametric bootstrap [10] with the following equations
i. This
u ¼1
B?
X
1
B
i¼0
^ui
(18)
Cu,Boot¼
B ? 1?
X
B
i¼0
(^ui? u)(^ui? u)T
(19)
As only a sample number of B ! 1 leads to a correct
calculation of the parameter variance Cu, a compromise
between the accuracy and computational effort has to be
found. Typically, 1000 to 10 000 samples are necessary to
obtain reasonably accurate results. As each sample involves a
PI step, and as a Bootstrap only returns a single evaluation of
the objective function for OED, the Bootstrap method is
prohibitively expensive for the use in the framework of OED.
2.5 Sigma points
Julier and Uhlmann [15] suggested the use of the so-called
SPs in order to determine the mean and covariance of a
random variable h [ Rlfrom the mean and covariance of
a random variable j [ Rf, where h is related to j by the
nonlinear mapping
h ¼ g(j) (20)
They showed that an accurate estimation of E[h] and Chcan
be obtained from (2 ? f þ 1) evaluations of g(?) for the
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(2 ? f þ 1) deliberately chosen samples of j (details of the
method will be given below). Although the method was
originally developed for applications in the area of nonlinear
filtering, known as the unscented Kalman filtering [16], it
can also be useful for OED problems, as will be shown in
the following. To simplify the discussion, it is assumed that
the measurement noise is additive, that is
y ¼ h(x) þ v
E[v] ¼ 0
E[v ? vT] ¼ Cy
(21)
(22)
(23)
although the method is not restricted to this case.
Then the PI problem can be formally written in the form
of (20), where
† j is the vector of all available measurement data. If K is the
number of measurement time points, j has the dimension
f ¼ m ? K and is given by
j ¼ [y1(t1), ...,y1(tK), y2(t1), ..., y2(tK), ...,
ym(t1), ...,ym(tK)]T
† g(?) stands for the complete PI process, that is, the
determination of a parameter vector^u that minimises the
objective function equation (4) by numerical optimisation;
† h is the resulting vector of estimated model parameters,
that is, h ¼^u.
Byusingthisanalogy,theSPmethodbyJulierandUhlmanncan
beapplieddirectlytocalculatethemeanandthecovariancesof^u.
For a measurement vector y with the dimension m and K
measurement time points, the covariance matrix of the
measurement noise Cyis used to determine 2mK þ 1 vectors
of ei
e0¼ 0
ei¼ þ
ei¼ ?
(24)
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
(mK þ l)
(mK þ l)
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p
ð
ð
ffiffiffiffi
C
C
ffiffiffiffi
p
p
yÞi;
i ¼ 1, ...,mK
i ¼ mK þ 1, ...,2mK
(25)
p
yÞi;
(26)
where ð
disturbances ei, which represent the statistics of the
measurement, generate a well-defined set of measurement
samples
ffiffiffiffi
C
p
yÞiis the ith column of the matrix square root. The
ys
i¼ h(x, u) þ ei;
i ¼ 0, ...,2mK
(27)
These samples are used to identify 2mK þ 1 parameter
vectors^uiwith the mean u and the covariance matrix Cu,SP,
u ¼
X
X
2mK
i¼0
2mK
wm
i^ui
(28)
Cu,SP¼
i¼0
wc
i(^ui? u)(^ui? u)T
(29)
where the weights wiare given by
wm
0¼
l
mK þ l
l
mK þ lþ 1 ? a2þ b
i¼ wc
(30)
wc
0¼
(31)
wm
i¼
1
2 ? (mK þ l);
i ¼ 1, ..., 2mK
(32)
with l ¼ a2? (mK þ k) ? mK and a, b and k are scaling
parameters. The meaning and the influence of a, b, k are
explained in [15]. In addition, the estimated mean u and the
initial parameter vector^uinienable the definition of a bias
Bi ¼ u ?^uini
(33)
that leads to the following approximation of the parameter
covariance matrix [11]
E (^u ? u)(^u ? u)T
hi
¼ BiBiTþ Cu,SP
(34)
Forthesakeofsimplicity,theinitialparametervectorissettothe
actualreferencevalue:^uini¼ u.Then,thebiasdefinedin(33)is
identical to the deviation of the estimates’ mean value from the
reference value. In the general case, the reference values of the
parameters are, of course, unknown a priori. Nevertheless,
the SP method can still be used to assess if the parameter
estimation method produces a bias or not. In order to do this,
^uiniis set to a first estimation^u0, which is usually different
from u. The SPs are grouped around this first estimate, and an
expectation u of the PI is computed from (28). If u =^u0,
then the identification method is found to produce a bias, and
the estimate of the bias is
^Bi ¼ u ?^u0
(35)
Compared with the FIM and the Bootstrap, the SP method
has a number of advantages that make it highly attractive for
OED
† One not only obtains a lower bound of Cuas from the
FIM, but an estimate of parameters that is highly accurate
as long the measurement noise is not too large.
† While the FIM postulates an unbiased parameter
estimator, the SP method also gives reliable information on
the estimated mean value of^u. Therefore in the framework
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of OED, it becomes possible not only to minimise the
covariance but also the bias of the estimated parameters.
† The SP method does not assume some idealised PI
procedure, but works with the numerical method that is
actually applied to the identification problem. Therefore all
kinds of imperfections and special properties of the
numerical optimiser are taken into account.
† In contrast to the FIM, the SP method does not require
the calculation of gradients or Jacobians for the sensitivities.
Therefore it is applicable to a broad class of models,
including for example, Monte Carlo models.
† The implementation of the SP method is very easy, the
parallelisation is straightforward.
† As the SPs are not chosen randomly but deliberately, the
number of required samples will usually be much smaller
than for the Bootstrap. In addition, the necessary number
of samples is clearly defined and not a matter of guessing,
as it is to some extent for the Bootstrap method.
3
3.1 Substrate uptake in a bio-reactor
Applications
The application of OED is demonstrated by a simple
unstructured growth model for a continuous stirred tank bio-
reactor. Under the assumption that the inlet flow qin[lh?1]
and the outlet flow qout[lh?1] of the reactor are equal (qin¼
qout¼: q[lh?1]), the dilution rate is defined as D ¼ q=V
[h?1] and the material balance equations for concentration of
biomass cx[gl?1] and substrate cs[gl?1] can be written as
_ cx¼ m ? cx? D ? cx
_ cs¼ ?1
Yxjs
(36)
? m ? cxþ (cs,in? cs) ? D
(37)
wherethespecificgrowthratem isdeterminedbyaMichaelis–
Menten kinetics
m ¼mm? cs
Ksþ cs
(38)
The simplicity of this unstructured model does not provide
a deeper insight into biological mechanisms, but it seems
appropriateto demonstrate
determining and minimising the parameter covariance matrix
for models that are nonlinear in their parameters. In total, the
model has three parameters: (i) Yxjs – the yield factor
describes how much biomass is produced by the uptake of a
certain amount of substrate; (ii) mm– the maximum growth
rate is the upper limit of the growth rate m and (iii) Ks–
the substrate affinity constant represents the substrate
concentration at which the specific growth rate is half its
maximum value. For reasons of simplification, the following
assumptions are made: (i) only the concentration of biomass
the typicalproblems of
is measurable y(tk) ¼ cx(tk) and (ii) Yxjsis known from
literature; consequently, only the parameters of the Michaelis–
Menten kinetics, mmand Ks, have to be identified, which
results in a two-dimensional parameter space.
3.1.1 Parameter identifiability: Before starting the PI,
it seems reasonable to check if the unknown parameters
can, in principle, be determined uniquely from the available
measurement information. Here, available measurement
information means the measurement data y(t) taken over some
finite time interval t [ 0, T
½
[2] – the measurement data y(0) at time zero plus the time
derivatives _ y(0), € y(0), ... up to some arbitrary order.
?, or – what is equivalent to this
To simplify the analysis, it is assumed in the following that
the initial substrate concentration cs(0) is known. One obtains
from the model equations (36) and (37)
y(0) ¼ cx0
_ y(0) ¼ _ cx(0) ¼ (m ? D) ? cx0¼ (m ? D) ? y(0)
€ y(0) ¼ (m ? D)_ y(0) þ@m
(39)
(40)
@cs
? _ cs(0) ? y
(41)
¼_ y(0)2
y(0)þ y(0) ?@m
@cs
??1
Yxjs
? m ? y þ D ? (cs,in? cs)
!
(42)
Equations (40) and (42) can be rearranged to
m??t¼0¼ D þ_ y(0)
@m
@cs
t¼0
y(0)
????
¼
€ y(0) ? ( _ y(0)2=y(0))
y(0) ? ( ? (1=Yxjs) ? (D ? y(0)
þ _ y(0)) þ D ? (cs,in? cs(0)))
¼: m0(0)
In other words, it is possible to express m??t¼0and (@m=
From the definition of m, it follows immediately that also
Ksand mmcan be expressed uniquely by known quantities
in the following way
@cs)jt¼0by the known quantities y(0), _ y(0), € y(0) and cs(0).
Ks¼
cs(0)2
(m(0)=m0(0)) ? cs(0)
mm¼m(0) ? (Ksþ cs(0))
(43)
cs(0)
(44)
This shows that it is – at least in principle for the case of
perfect measurement – possible to identify Ksand mm
from the available measurement information.
3.1.2 Confidenceintervalsofestimatedparameters:
The results of the previous section are based on a theoretical
concept and hence the parameter identifiability is only a
necessary condition for successful PI. In reality, one has to
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cope with a finite number of measurements taken at discrete
time points and corrupted by noise that leads to an
uncertainty in the estimated parameters. To assess the
quality of the identification process, it is important to
determine this uncertainty in terms of the covariance
matrix Cu. In Section 2, three numerical methods have
been discussed, two more traditional ones (FIM and
Bootstrap) and the SP method as a novel approach. The
three methods will be now compared by applying them to
the simple bio-reactor example. The following assumptions
are made:
† the measurement of the biomass concentration is taken at
three time points tk¼ [0:5 1:0 1:5] h;
† artificial measurement data are used, which are obtained
from a reference simulation and corrupted by normal
distributed noise.
The reference values of the parameters are Ks¼ 2 and
mm¼ 5. The variance of the measurement noise is taken as
Cy¼ 10?4[g2l?2]. This extremely small measurement error
is chosen deliberately to demonstrate that the used methods
provide different results in spite of almost unrealistically
precise measurements. The
MATLAB, using the time integrator ODE15s and the
optimiser LSQNONLIN, which uses the ‘Levenberg–
Marquardt’ method for optimisation. All optimisations
start with the reference values.
computationisdone in
In a first step, the Bootstrap method is used. An intrinsic
problem of the bootstrap approach is the correct choice of the
number of samples, that is, the question which number of
samples B is sufficient to obtain accurate results for the
mean value u of the estimated parameters as well as for
their covariance matrix Cu.
To overcome this, the Bootstrap is done repeatedly with an
increasing sample number B, so the characteristic values of
the parameter statistics are a function of B (Fig. 2).
Apparently, the Bootstrap needs more than 2000 samples
until a convergence is detectable. Only if the number of
samples is increased to B ¼ 10 000, the values of the mean
and the covariances tend to stabilise. One can also see
from Fig. 2 that the PI is slightly biased. The estimated
mean values of mmand Ksare slightly different from the
reference values.
In a second step, the SP method is tested. For a scalar
measurement (m ¼ 1) and three measurement time points
(K ¼ 3), one needs 2mK þ 1 ¼ 7 SPs, that is, 7 samples.
Compared with the 10 000 samples of the Bootstrap, this
is a tremendous reduction of the computational effort.
Nevertheless, both methods produce nearly the same
results. This indicates that the SP method works very well
and is highly effective in this case.
In a third step, the FIM, evaluated at the reference values,
is applied to the problem. As discussed in Section 2, the FIM
is based on the assumption of an unbiased parameter
estimation. Therefore it only provides information on the
covariancesbut not on the mean values of the estimated
parameters. A comparison of the 95% confidence intervals
determined by the three methods is given in Fig. 3.
It is evident that the FIM understimates the parameter
uncertainties strongly, especially the variance of Ks.
The results of the three methods are summarised in
Table 1. Obviously, the SP method and the Bootstrap with
10 000 samples agree well, whereas the FIM only gives a
poor estimate for the lower bound of the covariances, but
no estimate at all of the mean values.
Figure 2 Values of the mean and the variances of mmand Ksdetermined by SP and by Bootstrap, where the result of the
latter strongly depends on the sample number B
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Both effects, the underestimation of the parameter
uncertainties and the neglecting of the bias by the FIM
increase with a growing measurement error (Fig. 4). The
Bootstrap approach and the SP method provide almost
the same results for smaller measurement errors (Cy,
0:082[g2l?2]). For larger measurement errors, the Bootstrap
approach encounters numerical problems as the optimiser
fails for a number of samples. This is another point in
favour of the SP method, which works reliably also for
large measurement errors.
It is obvious that, at least for this example, the SP method
clearly offers by far the best compromise between the accuracy
and computational effort. The determined covariances can
be used to investigate the correlation between the two
parameters. All three methods calculate nearly the same
correlation coefficient r, defined as
rij¼
Cui,j
ffiffiffiffiffiffiffiffiffi
Cui,i
q
?
ffiffiffiffiffiffiffiffiffi
Cuj,j
q
(45)
The correlation coefficient is very close to unity (Table 1)
which indicates a very strong correlation between the two
parameters. This is clearly an unsatisfactory situation, but
agrees with experimental findings that it is very hard
to estimate mmand Ksaccurately from measurement data
[1, 17]. The question arises if it is possible to find more
favourableexperimental conditions for
estimation. This leads to the problem of OED, which is
discussed in the next section.
the parameter
3.1.3 Optimal experimental design:
covariance matrix Cuis known, the problem of OED, that is,
the reduction of the parameter uncertainties by varying design
parameters u(t), can be addressed. The inlet flow qin(t) is
defined as the design variable, so the objective is to find an
optimal trajectory of qin(t), which maximises the measurement
information content and minimises the parameter variances,
respectively. This numerical optimisation problem can be
solved, using one of the above-mentioned optimality criteria.
As the correlation of mmand KS makes the estimation
difficult, the E?-criterion (10) is used to reduce both, the
Oncethe
Figure 3 95% confidence interval (u¯i+ 2.p(C^u;ii)) of mmand Ksdetermined by FIM, SP and Bootstrap with a sample number
of B ¼ 10 000
Table 1 Characteristic values of the parameter statistic before the OED
FIM SP Bootstrap
Before OED
s2
Ks
0.1745 ? 1023
0.2105 ? 1023
–
0.1694 ? 1021
0.1076 ? 1021
2.0028
0.1689 ? 1021
0.1070 ? 1021
2.0023
s2
mm
E[Ks]
E[mm]– 5.0022 5.0018
r
0.99590.99660.9965
E?-criterion4.9118 ? 102
6.1317 ? 102
6.0063 ? 102
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correlation and the parameter uncertainties. The theoretical
minimum of FE?(Cu) is 1, that is, lmax(Cu) ¼ lmin(Cu),
suggesting that there is no correlation between the parameters.
The inlet flow is defined as the following linear function
q(t) ¼ a þ b(t ? c) (46)
which is a veryefficient inlet profile for the improvement of the
parameteraccuracyof such a simple unstructured growth model
[17]. Now the OED problem consists in the optimal choice of
theparametersa,bandc,whichminimisestheobjectivefunction
FE?(Cu). The covariance Cuis computed either from the FIM
or by the SP method. The Bootstrap is not used for OED, as it
would require extremely long computational times. As the
covariances computed by FIM and SP differ, the two
methods provide different conditions for a new experiment
(Table 2). Consequently, the inlet flow qin(t) and the
corresponding growth rate m(t) (Fig. 5) differ. The designed
experiments start with a higher inlet flow, leading to an
increased substrate concentration cS. As a result, the growth
rate m(cS(t)) is close to mm, which should provide a better
estimation of the maximum growth rate, but without a benefit
Figure 4 95% confidence intervals and expected values of the estimated parameters with respect to an increasing
measurement error, that is, standard derivation (std)
Table 2 Parameter of the inlet flow function qin(t) before
(INI) and after the OED for both methods (FIM and SPs)
abc
INI0.100 0.0000.000
FIM0.925 1.257 0.992
SP0.666 3.0890.867
Figure 5 Inlet flow qinand growth rate m of the unoptimised (dark grey) and the two designed experiments: FIM (light grey)
and SP (black)
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ofKS.TheSPmethodsuggestsanearlierandsteeperincreaseof
the inlet flow, which mayallowa more precise estimation of the
substrate limiting constant. To judge which of the two
experiments suggested by the FIM and by the SP method is
really the best, a Bootstrap is made for both experiments. The
resulting scatter plot (Fig. 6) with 2000 samples per newly
designed experiment shows that the SP method generates
a smaller and more roundish point cloud of^u, which is
equivalent to a reduction of the parameter uncertainties and
their correlation. Especially, the variance of KSin the SP-
designed experiment is smaller than in the FIM-designed
experiment. The main reason for this seems to be that the
inverse FIM underestimates the variance of Ks. Therefore in
the FIM-based OED, a high accuracy of the Ksestimate is
implied, and no effort is made to increase it further.
Furthermore, the scatter plot provides characteristic values of
the parameter statistics (Table 3), which correspond very well
with the results of the SP method. This demonstrates again
the reliability of the SP method.
To illustrate that the process of PI does not end up in a
local minimum, the contour plot of the sum of the
weighted squared errors (4) is analysed (Fig. 7). In all three
cases, that is, the unoptimised, the FIM optimised and the
SP optimised experiment, the cost functions have only one
global minimum. Therefore the application of a local
optimiser does not pose a problem. Furthermore, the
contour plots can also be used toassess the results of the
OED independent of the used optimiser. As the SP
method provides the most appropriate contour, that is,
tight and roundish, the SP-designed experiment not only
have a benefit on the Levenberg–Marquardt optimiser but
also on any other minimum search algorithm.
3.1.4 Confidence regions of the dynamic states:
An additional advantage of the SP method is that it can
also be used to investigate the influence of the parameter
uncertainties on the simulation results. To do this, one has
to use for j in (20) the estimated parameter vector^u, g(?)
stands for the solution of the ODE system and h
represents the state vector. The possibility to determine a
confidence region of the states xiis especially interesting in
the field of observation and control, and also a coming up
in systems biology [18]. Furthermore, the 95% confidence
regions of the dynamic states (Fig. 8) obviously explain the
benefit of the OED.
After a successful demonstration of the potential of the SP
method, the OED is realised for a more complex model in
the next section.
3.2 Incorporation of intracellular
components
To demonstratethat the SP method is also applicable to more
complex models, the simple bio-reactor model is extended by
three intracellular components (cm,1cm,2cm,3[gl?1]). The
corresponding balance equations can be formulated as
_ cx¼ m ? cx? q ? cx
_ cs¼ ?1
Yxjs
_ cm,1¼ r1? r2? m ? cm,1
_ cm,2¼ r2? r3? m ? cm,1
_ cm,3¼ r3? r4? m ? cm,1
(47)
? r1? cxþ (cs,in? cs) ? q
(48)
(49)
(50)
(51)
Figure 6 Scatter diagram of the optimised parameter estimation
The SP-designed experiment provides a tighter and more roundish point cloud (black dots) than the use of FIM (grey dots), indicating that
the SP enable a higher information content and an improved parameter estimation, respectively
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Page 10
wheretheratesareasetofmassactionandMichaelis–Menten
kinetics
r1¼mm,1? cs
Ks,1þ cs
r2¼mm,2? cm,1
Ks,2þ cm,1
r3¼mm,3? cn
Kn
(52)
(53)
m,2
s,3þ cn
m,2
(54)
r4¼ mm,4? cm,3
(55)
Consequently,
increases from two to eight u ¼ [mm,1
Ks,2
mm,3
Ks,3
mm,4
thenumberof unknown parameters
Ks,1
mm,2
n]
Following assumptions about the measurement information
are made:
† the concentration of every component is measured;
† measurement
tk¼ [0:2 0:8 1:4 2:0]h;
samples aretakenat fourtimes
† artificial measurement data are used, which are obtained
from a reference simulation and corrupted by normal
distributed noise;
† the variances of the measurement noise are assumed as
Cy,ii¼ 0:12[g2l?2] for all components.
For the measurement vector y(m ¼ 5) and the four
measurement time points (K ¼ 4), the number of SPs
increases to 2mK þ 1 ¼ 41. Compared with the at least
1000 samples of a Bootstrap approach, it is still a
considerable reduction of computational effort.
3.2.1 Determination and reduction of the
parameter uncertainties: After a first PI, the statistics
of the parameters are determined. As the Bootstrap
approach requires too much computational time, only the
FIM and the SP method are applied to the extended model.
In Fig. 9, the 95% confidence regions of the estimated
parameters^u are shown. Obviously, the assumption of an
unbiased estimator is not met; furthermore, the confidence
regions do not agree, that is, the covariance matrix Cu
calculated by the FIM and SP method are different.
The previous sections pointed out clearly that the inverse of
the FIM provides a relative unrealistic approximation of the
actual parameter statistics, hence the OED is realised with
the SP method in the next step. Once again the inlet flow
q(t) is defined as the design variable, to minimise the
parameter uncertainties. To show that the accuracy of the
parameter estimation can be improved with a minimal
effort, q(t) is assumed as a constant q(t) ¼ qopt.
Using the A-optimal design criteria (6) for the OED
process, the SP method provides qopt¼ 0:336[lh?1] as the
optimal choice of the design variable. A new determination
of the parameter covariance matrix via the SP method for
the optimally designed experiment indicates (Fig. 10) that
it is not possible to reduce the parameter uncertainties of
Table 3 Characteristic values of the parameter statistics after the OED of the FIM-designed and the SP-designed experiment
FIM SPScatter plot
FIM-designed experiment
s2
Ks
0.2667 ? 1025
0.0453 ? 1025
–
0.9842 ? 1023
0.0609 ? 1023
2.0026
0.9891 ? 1023
0.0611 ? 1023
2.0020
s2
mm
E[Ks]
E[mm]– 5.0007 5.0005
r
0.96940.9967 0.9971
E?-criterion1.3159 ? 102
2.9919 ? 103
3.1406 ? 103
SP-designed experiment
s2
Ks
0.5122 ? 1025
0.1410 ? 10025
–
0.2087 ? 1023
0.0462 ? 1023
1.9990
0.2130 ? 1025
0.0452 ? 1025
1.9999
s2
mm
E[Ks]
E[mm]–4.9996 4.9999
r
0.9891 0.82540.8584
E?-criterion 2.7148 ? 102
24.535524.2887
For both results, the values are determined via the FIM, SP method and the Bootstrap approach (scatter plot)
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all parameters in one step considerably, for example, the
uncertainty of the parameter n is increased. This result is a
well-known problem in the field of OED, which can
be solved by estimating only the ‘important’ subset
ui# u of the unknown parameters, whereby the uu¼ unui
parameters are assumed to be known and not part of the
estimation [19].
Nevertheless, the global benefit of the OED to the dynamic
states x(t) is much bigger than the confidence regions of the
Figure 7 Contour plots of the parameter estimator cost function (sum of the weighted squared errors) for the unoptimised,
the FIM-optimised and the SP-optimised experiment
Figure 8 95% confidence regions of the dynamic states: grey for the unoptimised and black for the SP-optimised parameter
estimation
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estimated parameter (Fig. 9) suggest. The uncertainties of all
dynamic states are reduced obviously (Fig. 11). This effect
agrees well with the assumption that only a few linear
combinations of the unknown parameters determine the
qualitative behaviour of the a model [19, 20]. The objective of
OED, obtaining models with an increased predictive power,
could be achieved with a minimal effort, for example,
minimal cost and minimal technical equipment.
Figure 9 95% confidence intervals of the estimated parameters determined by FIM and the SP method
Figure 10 95% confidence intervals of the estimated parameters before and after the OED
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4 Conclusions
In this paper, we have presented a general, systematic
procedure to calculate and minimise the parameter covariance
matrix Cufor models that are nonlinear with respect to their
parameters. Starting with the determination of Cuto assess
the quality of a PI, the disadvantages of the traditionally
used FIM method become clear. The Bootstrap approach
and the SP method are able to approximate Cuin a much
more realistic way.
If the FIM is used for an OED, the process is designed
under assumptions that are not met by many dynamical
systems, leading to a sub-optimal choice of the design
variables. Owing to the fact that the computational effort
of the Bootstrap approach prohibits its application to the
OED process, this leaves the SP method as an attractive
alternative for determining and minimising the parameter
uncertainties. Further, confidence regions of the dynamic
states can beobtainedvia
makes the benefit of OED obvious and enables novel
approaches of cost functions to improve the predictive
power of models.
theSP method.This
5 Acknowledgment
We
Forschungszentrum fu ¨r Dynamische Systeme in Biomedizin
und Prozesstechnik, Sachsen-Anhalt.
gratefully acknowledge the financialsupportby
6 References
[1]
microbial growth models incorporating Michaelis–Menten
type nonlinearities’, Mathe. Biosci., 1982, 62, pp. 23–43
HOLMBERG A.: ‘On the practical identifiability of
[2]
(Springer, 1997)
WALTER P.L.: ‘Identification of parametric models’
[3]
design in systems biology, based on parameter sensititivity
analysis using a monte carlo method: a case study for the
TNFa-mediated NF-k B signal transduction pathway’,
Simulation, 2003, 79, pp. 726–739
WOLKENHAUER O., KOLCH W., CHO K.H., SHIN S.Y.: ‘Experimental
[4]
bootstrap method for quantifying parameter confidence
intervals in dynamical systems’, Metab. Eng., 2006, 8,
pp. 447–455
JOSHI M., SEIDEL-MORGENSTERN A., KREMLING A.: ‘Exploiting the
[5]
some related control problems’, Automatica, 2008, 44,
pp. 303–325
PRONZATOL.: ‘Optimal experimental design and
[6]
experiments for parameter precision: state of the art’,
Chem. Eng. Sci., 2008, 63, pp. 4846–4872
FRANCESCHINI G., MACHIETTO S.: ‘Model-based design of
[7]
estimation theory’ (Prentice Hall PTR, 1993)
KAY S.M.: ‘Fundamentals of statistical signal processing:
Figure 11 95% confidence regions of the dynamic states before and after the OED
22
& The Institution of Engineering and Technology 2009
IET Syst. Biol., 2009, Vol. 3, Iss. 1, pp. 10–23
doi:10.1049/iet-syb:20080094
www.ietdl.org
Page 14
[8]
distillation processes’ (Otto-von-Guericke-Universita ¨t
Magdeburg, 2007)
GANGADWALA J.: ‘Optimal design of combined reaction
[9]
N., NICOLAI B.M.: ‘Sensitivity analysis of microbial growth
parameter distribution with respect to data quality and
quantity by using Monte Carlo analysis’, Math. Comput.
Simul., 2003, 65, pp. 231–243
IMPE J.F.V., POSCHET F., BERNAERTS K., GEERARD A.H., SCHEERLINCK
[10] EFRON B., TIBSHIRANI R.J.: ‘An introduction to the bootstrap’
(Chapman & Hall, 1993)
[11] EMERY A.F., NENAROKOMOV A.V.: ‘Optimal experiment
design’, Meas. Sci. Technol., 1998, 9, pp. 864–876
[12] VERSYCK K.J., IMPE J.F.V.: ‘Feed rate optimization for fed-
batch bioreactors: from optimal process performance to
optimal parameter estimation’, Chem. Eng. Commun.,
1999, 172, (1), pp. 107–127
[13] LINDNER
for optimal
kinetic
Fisher information matrix’, J. Theor. Biol., 2006, 238,
pp. 111–123
P.F.O.,
parameter estimation
processbased
HITZMANN B.: ‘Experimental design
of an
the analysis
enzyme
of on the
[14] BALSA-CANTO E., RODRIGUEZ-FERNANDEZ M., BANGA J.R.: ‘Optimal
design of dynamic experiments for improved estimation of
kinetic parameters of thermal degradation’, J. Food Eng.,
2007, 82, pp. 178–188
[15] JULIER
approximating nonlinear transformation of probability
distributions, 1996’. Available from http://www.robots.ox.
ac.uk/?siju/work/work.html
S.,
UHLMANNJ.: ‘Ageneralmethod for
[16] QUACH M., BRUNEL N., D’ALCHE´BUC F.: ‘Estimating parameter
and hidden variables in non-linear state-space models
based on ODEs for biological networks inference’,
Bioinformatics, 2007, 23, pp. 3209–3216
[17] BALTES M., SCHEIDER R., STURM C., REUSS M.: ‘Optimal
experimental design for parameter estimation in unstructured
growth models’, Biotechnol. Prog., 1994, 10, pp. 480–488
[18] LIEBERMEISTER W., KLIPP E.: ‘Biochemical networks with
uncertain parameters’, IEE Proc. Syst. Biol., 2005, 152,
pp. 97–107
[19] GUTENKUNST R.N., WATERFALL J.J., CASEY F.P., BROWN K.S., MYERS C.R.,
SETHNA J.P.: ‘Universally sloppy parameter sensitivities in systems
biology models’, PLoS Comput. Biol., 2007, 3, pp. 1871–1878
[20] CASEY F.P., BAIRD D., FENG Q., GUTENKUNST R.N., WATERFALL J.J.,
MYERS C.R., ET AL.: ‘Optimal experimental design in an
epidermal growth factor receptor signalling and down-
regulation model’, IET Syst. Biol., 2007, 1, pp. 190–202
IET Syst. Biol., 2009, Vol. 3, Iss. 1, pp. 10–23
doi:10.1049/iet-syb:20080094
23
& The Institution of Engineering and Technology 2009
www.ietdl.org
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