# Optimal experimental design with the sigma point method.

**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.

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- Computers & Chemical Engineering 01/2014; 71:415-425. · 2.09 Impact Factor
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**ABSTRACT:**State-of-the-art formulations of optimal experiment design problems are typically based on a design criterion which allows us to optimize a scalar map of the predicted variance–covariance matrix of the parameter estimate. Famous examples for such scalar objectives are the A-criterion, the E-criterion, or the D-criterion, which aim at minimizing the trace, maximum eigenvalue, or determinant of the variance–covariance matrix. In this paper, we propose a different way of deriving an economic design criterion for the optimal experiment design. Here, the corresponding analysis is based on the assumption that our ultimate goal is to solve an optimization problem with a given economic objective that depends on uncertain parameters, which have to be estimated by the experiment. We illustrate the approach by studying a fedbatch bioreactor.Automatica. 11/2014; - SourceAvailable from: René Schenkendorf05/2014, Degree: Dr.-Ing.

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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.

1Introduction

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,

thedeterminationofunknown 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

kindofrealisationhasan

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.

increasedcomputational

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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

appropriatetodemonstrate

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 typicalproblemsof

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

precisemeasurements.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.

computation isdonein

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

favourable experimental 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 SPBootstrap

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]–5.00225.0018

r

0.9959 0.99660.9965

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

INI 0.1000.0000.000

FIM0.925 1.257 0.992

SP0.666 3.089 0.867

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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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

the numberof unknownparameters

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;

samplesaretakenatfour times

† 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

FIMSPScatter 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

0.96940.99670.9971

E?-criterion 1.3159 ? 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]– 4.99964.9999

r

0.9891 0.82540.8584

E?-criterion 2.7148 ? 102

24.5355 24.2887

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 canbe obtainedvia

makes the benefit of OED obvious and enables novel

approaches of cost functions to improve the predictive

power of models.

theSP method. This

5Acknowledgment

We

Forschungszentrum fu ¨r Dynamische Systeme in Biomedizin

und Prozesstechnik, Sachsen-Anhalt.

gratefullyacknowledge thefinancialsupportby

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