Directory UMM :Data Elmu:jurnal:A:Agricultural Economics:Vol22.Issue2.Mar2000:

Agricultural Systems 66 (2000) 1±15
www.elsevier.com/locate/agsy

Summing up dynamics: modelling biological
processes in variable temperature scenarios
L.M.M. Tijskens *, F. Verdenius
ATO, PO Box 17, 6706 AA Wageningen, The Netherlands
Received 15 December 1999; received in revised form 31 March 2000; accepted 10 May 2000

Abstract
The interest of modelling biological processes with dynamically changing external conditions (temperature, relative humidity, gas conditions) increases. Several modelling approaches
are currently available. Among them are approaches like modelling under standard conditions, temperature sum models and dynamic modelling. While the ®rst two approaches
require huge simpli®cations that endanger the applicability of the results, the latter requires a
substantial modelling and computational e€ort. In this paper the often very successful method
of temperature sum is improved and enhanced to re¯ect fundamental insights in biochemical
processes. Knowing that reaction rates depend on temperature according to Arrhenius' law, a
rate sum calculation for each active process is proposed. While the temperature sum approach
is in practice restricted to polynomial models, the rate sum approach allows the building and
application of more fundamental and process-oriented models. The method is computationally feasible. Model calculations on simulated data show that this approach is at least
equivalent to existing approaches, and often outperforms them in terms of statistical ®t (R2adj
of over 90%, and often 99.5%). Moreover, it has the major advantage of estimating parameters that have an interpretation in the biochemical reality. Another major advantage is that

all the normal rules, techniques and procedures of statistics remain applicable. # 2000
Elsevier Science Ltd. All rights reserved.
Keywords: Dynamic temperature scenarios; Temperature sum; Rate sum; Dynamic modelling; Fundamental model formulation; Non-linear regression analysis

* Corresponding author. Tel.: +31-317-475-303; fax: +31-317-475-347.
E-mail address: l.m.m.tijskens@ato.wag- ur.nl (L.M.M. Tijskens).
0308-521X/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved.
PII: S0308-521X(00)00027-5

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L.M.M. Tijskens, F. Verdenius / Agricultural Systems 66 (2000) 1±15

Nomenclature
Variables
C
Concentration of compound or property
Ea
Energy of activation
Enz

Enzyme activity
k
Rate constant
kS
Rate integral or rate sum
P
Concentration of some product of reaction
R
Universal gas constant
Percentage variance accounted for
R2adj
T
Temperature
t
Time
TS
Temperature sum

Indices
0

1,2,3
d
i,j
n
na
p
o€set
ref
s

Initial value at t=0
Occurrences of concentrations or processes
Denaturation
Running array indicators
Number of observations
Not active
Process number indicator
Applied o€set
At reference temperature
Substrate

1. Introduction
Developing models for use in agriculture frequently involves working with dynamically changing external conditions. This mainly concerns temperature that varies
during the growing of produce, as well as during the subsequent storage and handling in the entire food chain. Other external factors that are often dynamic are relative humidity and (solar) radiation. In scienti®c studies, the experimental conditions
are kept as constant as possible to avoid the troublesome dynamics in interpreting
and modelling the results. Especially in agricultural research, control of external
factors like temperature, relative humidity and radiation intensity is often impossible. This is not only the case during production in open ®eld, but also in many
parts of distribution chains. Consequently, these external factors vary too much to
consider them as constant. The dynamics of the environment have to be approximated to include their e€ects in models and data analysis. Existing approaches for
these approximations either make radical simpli®cations, such as assuming mean

L.M.M. Tijskens, F. Verdenius / Agricultural Systems 66 (2000) 1±15

3

values for external conditions, or result in hard-to-interpret results, as obtained in
polynomial or purely statistical modelling.
In this paper a new and fundamental approach is developed and described to deal
with dynamic temperature scenarios. The e€ect of temperature on rates of occurring
processes as described by Arrhenius' law is included, without harsh simpli®cations

or assumptions. The potential of its application is indicated with a few examples.

2. Describing the problem
In modelling the behaviour of products and processes, the primitive, most basic
functions usually applied are di€erential equations. Often analytical solutions can be
derived for these sets of di€erential equations for constant external conditions as,
for example, temperature. These analytical solutions can be used for statistical analysis of experimental data, obtained at constant conditions. And these are exactly
the conditions that applied statistics have promoted for experimental research to be
conducted. There is a large set of robust tools and analysis schemes from applied
statistics to facilitate this approach.
On many occasions, however, it is impossible to conduct experiments at constant
external conditions. Only in well-controlled production and storage facilities, e.g.
greenhouses, does the assumption of constant conditions hold. In outdoor farming,
however, control of temperature, rainfall or solar radiation is no option. Hence, in
practice, conditions are highly dynamic. Applicability of the derived model, functions and results is therefore very limited. To make at least an attempt to use these
valuable data several approaches are applied, all having some drawbacks on reliability or practical applicability.

3. Traditional approaches
3.1. Applying mean temperature
When the variation in temperature is not all too large, a possible and oftenapplied approach is to use the mean temperature during the period of experimenting, as a substitute for the real occurring temperature. In storage experiments, for

example, the temperature in controlled storage rooms does ¯uctuate around the set
point. The ¯uctuations are usually small with an order of magnitude of one to two
degrees. Also, the e€ect of these temperature ¯uctuations is normally considered less
important then the e€ect of product variance and measuring error. So, the temperature ¯uctuations are almost never taken into consideration in the statistical
analysis of the experimental results, and the classical analysis for constant conditions is applied. The reliability of the results is suciently high for a sensible and
useful interpretation.
In Eq. (1), an example is worked out for a simple ®rst-order or exponential process. The di€erential equation is:

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L.M.M. Tijskens, F. Verdenius / Agricultural Systems 66 (2000) 1±15

dC…t†
ˆ ÿk…T…t††  C…t†:
dt

…1†

At constant temperatures, T(t) is constant and the dynamic rate constant [notation k(T(t))] becomes independent from temperature and consequently from time
(notation k). Upon integration at constant conditions, the well-known exponential

function emerges:
C…t† ˆ C0  eÿkt :

…2†

The advantage of this approach is its ease of operation, and the reliable techniques
available for statistical analysis. The disadvantage is, of course, the fact that e€ects of
temperature ¯uctuations are cancelled out before their insigni®cance can be proven.
3.2. Applying dynamic modelling
When the di€erences in temperature are too large to neglect, the most sensible
approach is to use for statistical analysis directly the di€erential equations and the
e€ect of temperature on the parameters of the di€erential equations. In this case,
Eq. (1) would be used by combining a numerical integration with a statistical estimation procedure. The mathematics of the applied technique looks like:
…t
…t
k…T…t††  C…t†  dt:
…3†
dC…t† ˆ ÿ
tˆ0

tˆ0

This is the technique used in simulation programs and modelling languages. The
advantages of the technique are manifold. In the ®rst place, no further assumptions
are made to arrive at a statistical estimation of the value of the parameters, except
for the numerical integration. Good and reliable numerical integrators are amply
available nowadays. In the second place, there is no need to obtain an analytical
solution, which is often dicult or even impossible. The derived di€erential equations are used as is. In the third place, the sampling or measuring frequency does not
need to be at regular intervals. In the last place, dynamic experiments can provide a
very neat way to decrease the number of experiments to be conducted, and still
obtain the same amount of information.
There are, however, also drawbacks to the technique. Combinations of numerical
integrators with statistical robust optimisation algorithms are not readily available.
Most of the time a dedicated program has to be written (and tested) based on the
actual model and the actual data to be used. Examples of this technique can be
found in Verlinden et al. (1995, 1996, 1997). Some modelling languages are emerging
that are capable of performing parameter estimation on di€erential equations, e.g.
Modelmaker (Cherwell Scienti®c Ltd, Oxford, UK). However, these systems are not
yet capable of handling complex and heavily interactive models.
Another diculty with this method is that the information on temperature

dependence of the parameters of the considered model is utterly concealed. It is very

L.M.M. Tijskens, F. Verdenius / Agricultural Systems 66 (2000) 1±15

5

dicult to make clear where and how much information on temperature can be
obtained from a particular scenario. Designing scenarios to obtain the most ecient
way for conducting dynamic experiments is rapidly becoming a new subject of scienti®c study (Versyck et al., 1997a, b, 1998a, b, 1999; Bernaerts et al., 1998, 2000;
Van Impe, 1998; Van Impe et al., 1998).
3.3. Applying temperature sum
Another frequently used technique in black box or empirical modelling is the
application of the temperature sum over time. In calculating this temperature sum,
an o€set is sometimes considered to cut o€ temperatures that are known to be
without e€ect. For a series of collected temperatures Ti at a series of distinct times ti,
the temperature sum (TS) is at each time calculated as the sum of all temperatures
up to the actual time. This can mathematically be represented as:
TSi ˆ

i ÿ

X
jˆ1


ÿ


Tj ÿ Toffset  tj ÿ tjÿ1 for i from 1 to n:

…4†

This temperature sum is frequently used in modelling the observed behaviour of some
variable with second- or even higher-order polynomials or to apply combined time±
temperature information in existing models (De Visser, 1992; D' Antuono and Rossini, 1994; Grevsen, 1998). The temperature sum has been applied in combination
with machine learning approaches in an application to manage product inherent
variance (Verdenius, 1996). The temperature sum technique has proven to be reliable
and generically applicable in the pre- and postharvest chain of agricultural products.
Some reason has to exist why this empirical approach is so general applicable.
By applying this kind of transformation a new pseudo-variable (TS) is created that
contains all the information on all combinations of time and temperature occurring
in the experiments. The observed behaviour of the phenomena under study is now
explained based on this pseudo-variable applying mostly higher-order polynomials.

Of course, relations with processes that are more fundamental and fundamental laws
that could be applicable are lost as one does absolutely not know how this pseudovariable could possibly govern the occurring processes. The parameters of such
empirical models lack all physical, chemical, biochemical or physiological meaning.
Results are therefore hard to interpret, and impossible to translate to circumstances
di€erent from the experimental ones.

4. Alternative approach
At a closer inspection of Eq. (4), one can readily see that the temperature sum is a
crude approximation of the integral of the variable temperature over time:
…t
…T…t† ÿ Toffset †  dt:
…5†
TS…t† 
tˆ0

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L.M.M. Tijskens, F. Verdenius / Agricultural Systems 66 (2000) 1±15

The relation depicted in Eq. (5) also makes completely clear the importance of suf®cient sampling frequency, to make sure that the information contained in the
measured temperature data is sucient to approximate, by integration or summation, the real integral of temperature over time: is the applied temperature scenario
suciently described by the collected temperature data.
On closer comparison of Eqs. (3) and (5), it becomes evident where the drawbacks
of the application of temperature sum originate. In the theoretical relation (Eq. (3)),
the e€ect of temperature on the rate of processes is included before the integration,
whereas in the practical relation (Eq. (5)) that e€ect will be applied after integration
(or summation) of temperature over time.
As the processes occurring in living or even in dead biological produce are in most
cases chemical and biochemical processes, the rates of reaction will depend on temperature, presumably according to the law of Arrhenius:


kp ˆ kp;ref  e

Eap
R 

1
1
Tref ÿT

…6†

;

where p indicates the process involved. With dynamic temperature scenarios applied
to the biological material, this equation extends to:


kp …T…t†† ˆ kp;ref  e

Eap
R 

1
1
Tref ÿT…t†

…7†

:

Instead of integration over temperature directly, one should integrate for each
occurring process separately over the rate constant, as deduced in Eq. (3). This
results in a pseudo-variable kS, speci®c for each occurring process:
…t
…8†
kSp …t† ˆ kp …T…t††  dt:
0

When the di€erential equation for a ®rst-order exponential process (Eq. (1)) is
integrated with variable temperature included, one obtains (Pinheiro Torres and
Oliveira, 1999):
„t
ÿ k …T…t††dt
:
…9†
C…t† ˆ C0  e 0 p
The integral in the exponent is the same as the rate sum (Eq. (8)). For practical
application of this deduction, in the analytical solution (Eq. (2)) at constant external
conditions (of temperature), the term kpt can be replaced by the so-obtained
kSp …t†, which in turn can be approximated by a summation:
kSp;i 

i ÿÿ ÿ

X

ÿ



kp Tj ÿ kp …T1 †  tj ÿ tjÿ1
jˆ1

for i from 1 to n:

…10†

L.M.M. Tijskens, F. Verdenius / Agricultural Systems 66 (2000) 1±15

7

The term kp …T1 † constitutes a correction factor, similar to the trapezoidal rule for
approximating an integral with a summation. This correction factor ensures that kS
at the ®rst measuring point is zero.
The same fundamental approach to include the e€ect of temperature history in
model formulations based on application of Arrhenius' law was used by Wells and
Singh (1988). They, however, developed the idea further to estimate an e€ective temperature as a replacement for the mean temperature. In addition, Allen (1988) already
strongly reported that the mean of a function is de®nitely di€erent from the function
of the mean. In our case, the function is Arrhenius. Therefore, the rate at mean temperature is di€erent from the mean of the rates over the di€erent temperatures.
So, instead of using a general temperature sum in empirical modelling, one can
apply much more fundamental models using a rate sum speci®c for each process.
These fundamental models have a bearing on the processes occurring in the product.
Using these more fundamental models instead of empirical models, the understanding of what is going on in our products, the development of theoretical views,
the development of tools and systems increases very much. The scope of practical
application of models and theories is greatly enlarged (Tijskens et al., 1998a).

5. Theoretical examples
5.1. Materials, methods and data generation
All data used in this paper are simulated data based on models of simple reaction
mechanisms by numerical integration of the di€erential equations. All mathematical
developments of models and generation of simulation data were conducted with
MapleV release 4 (Waterloo Maple Inc., Waterloo, Canada). Statistical analysis was
conducted with the non-linear regression procedure of Genstat 5 (Rothamsted, UK).
For the next examples, data were generated by simulation using imaginary but
plausible values for the model parameters on three basic and simple models, which
occur in nature and are frequently applied in modelling of processes in agricultural
products. For each of the models, three temperature scenarios were applied, a simple
heating up and cooling down over 20 C, a sinusoidal change in temperature, to
mimic daily temperature ¯uctuations in the growing ®eld, and a simple scenario
to mimic the temperature in a food distribution chain. In Fig. 1 the three temperature scenarios are shown.
5.2. Simple ®rst-order degradation
Simple ®rst-order degradation is a process very frequently occurring in nature,
and applied commonly in modelling. The mathematical equations are already provided (Eqs. (1)±(9)) and used in the mathematical deduction. The mechanism is a
simple conversion:
k

C ! P:

…11†

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L.M.M. Tijskens, F. Verdenius / Agricultural Systems 66 (2000) 1±15

Fig. 1. Applied temperature scenarios.

For this process, the system of rate sum is fundamentally correct (Eqs. (8)±(10)) and
does not introduce another error except the approximation of an integral by a
summation.
In the analytical solution at constant temperature (Eq. (2)) the term kt may be
substituted by the rate sum as de®ned by Eqs. (8) and (10). This results in:
C…t† ˆ C0  eÿkS…t† :

…12†

This signi®es that this system can and may be applied at any time, provided the
information contained in the data, covering both the behaviour of the state variable
C, and the e€ect of temperature on the rate of reaction, is sucient to detect all
these e€ects. But that is a golden rule in any statistical analysis: if the data do not
contain information on a certain subject, no conclusions can be drawn from these
data on that subject.
Data were generated by numerical integration of Eq. (1), using the three dynamic
scenarios, and a value for the input parameters as shown in second column in Table
1. By applying the approximation of the rate sum kS(t) for each of the dynamic
scenarios instead of the factor kst the dynamics of the temperature scenarios can
be analysed using the standard statistical routine of non-linear regression analysis.
Table 1
Results of statistical data analysis with rate sums to incorporate dynamic scenariosa
Input

R2adj

Heat-cool

Daily

Distribution

All scenarios combined

Estimate S.E.

Estimate S.E.

Estimate S.E.

Estimate

100
100
99.9
C0
100 100.132
0.0791 100.925
0.117 98.6734
0.0555
ks,ref
0.06 0.060402 0.000288 0.060667 0.000172 0.055316 0.000428
Eas 10 0000 10 197.7
79
9758.1
84.2
9671
45.7

100
99.617
0.059385
9951.5

S.E.
0.0784
0.000104
24

a
Model experiments on ®rst-order reactions with exponential decay. S.E., standard error of estimate.
See Nomenclature for de®nition of other abbreviations.

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L.M.M. Tijskens, F. Verdenius / Agricultural Systems 66 (2000) 1±15

In Table 1 the statistical estimates of the input parameters are shown for each
dynamic scenario. For this simple exponential behaviour, the rate sum system is
fundamentally correct. The only approximation applied is found in the summation
of the rate, instead of integration. The results are therefore not surprisingly completely correct, as can be taken from the estimated values for the parameters, their
standard errors and the extremely high percentage variance accounted for (R2adj).
The three temperature scenarios were applied to the exponential model with the
same input values for the parameters (second column in Table 1). This means that
the data can be pooled together and analysed in their entirety, again using the
standard statistical procedures of non-linear regression. This technique of pooling
data takes advantage of the larger set of data, and the better description of the
temperature dependency in the three scenarios together. The results of the nonlinear regression analysis are shown in the last two columns of Table 1.
The ®rst-order exponential decay is also a part of the next examples on the activity
of a denaturating enzyme and on the consecutive reaction. The estimated value of
the parameters of the exponential enzyme denaturation analysed with non-linear
regression analysis using the exponential formulation (Eq. (12)) can be taken from
Table 2. The estimated values of the parameters of the exponential conversion of C1
into C2 analysed with the same technique are shown in Table 5. For each of the
three applied scenarios, the parameter used can be estimated very well. The temperature information in the distribution scenario is, however, too low to obtain a
really reliable estimate. This scenario contains only information in the region
around 20 C and around 2 C (Fig. 1). It is obvious that two data points are too few
for an allowed and reliable regression analysis.
5.3. Enzymatic conversion
The second example is the action exerted by a denaturating enzyme. This type of
behaviour frequently occurs when applying moderate heat treatment (e.g. blanching)
Table 2
Results of statistical data analysis with rate sums to incorporate dynamic scenariosa
Input

Heat-cool

Daily

Distribution

Estimate

S.E.

Estimate

S.E.

Estimate

S.E.

R2adj
Enz0
kd,ref
Ead

1
0.001
25 000

99.9
0.999974
0.001007
25 381

1.34E-05
1.29E-05
387

99.9
1.004558
0.010228
24 575

0.000983
0.000214
433

99.5
0.9978
0.009142
26 011

NA
NA
NA

R2adj
S0
ks,ref
Eas

100
0.1
5000

100
99.9344
0.100438
5135.7

0.0545
0.000177
19.6

100
99.904
0.09507
6360.5

0.138
0.000217
70.9

100
98.596
0.09916
4968.8

0.0911
0.00103
46.2

a
Model experiments on the exerted action of a denaturating enzyme. S.E., standard error of estimate;
NA, not available. See Nomencalture for de®nition of other abbreviations.

10

L.M.M. Tijskens, F. Verdenius / Agricultural Systems 66 (2000) 1±15

to agricultural products (Tijskens et al., 1999). The reaction mechanism is basically
a combination of a ®rst-order reaction and a second-order reaction as shown in
Eq. (13):
kd

Enz ! Enzna ;
ks

S ‡ Enz ! P ‡ Enz:

…13†

A set of di€erential equations can be derived based on the fundamental laws of
reaction kinetics. This set of di€erential equations is used to generate the simulated
data (Eq. (14)).
@Enz…t†
ˆ ÿkd  Enz…t†
@t
@S…t†
ˆ ÿks  Enz…t†  S…t†
@t
@P…t†
ˆ ks  Enz…t†  S…t†
@t

…14†

The analytical solution of this set of di€erential equations at constant temperatures
is:
Enz…t† ˆ Enz0  e…ÿkd t† ;


ks ÿ …ÿkd t†
Enz0 
 e
ÿ1
kd
:
S…t† ˆ S0  e

…15†

Data were generated by numerical integration of Eq. (14), using the three dynamic
scenarios, and a value for the input parameters as shown in second column in Table 2.
The behaviour of the substrate S at the three dynamic scenarios is shown in Fig. 2.
The factors kst and kdt can then again be approximated by the rate sums kSs
and kSd to include the dynamics of the temperature scenarios in the analysis. That

Fig. 2. Available substrate S for a denaturating enzyme for three dynamic scenarios (Eqs. (13) and (14)).

11

L.M.M. Tijskens, F. Verdenius / Agricultural Systems 66 (2000) 1±15

leaves the unresolved meaning of the rate constants ks and kd that occur in the
equation without being coupled to the time t (ks/kd). These parameters apparently
have a di€erent meaning in the rate sum approximation of the temperature dynamics than in the analytical solution and should re¯ect the dynamics of temperature. As
these rate `constants' in analytical solutions always occur in ratios, the time is cancelled out and they can be replaced by their respective rate sums: (kSs/kSd).
In Table 2 the statistical estimates of the input parameters are shown for each
dynamic scenario. In the ®rst analysis (top half of the table), the parameters for the
enzyme denaturation were estimated based on information contained in the generated data of Enz(t) only. In the second analysis (bottom half of the table), the
parameters of the enzymatic action were estimated, keeping the value of the parameters of the denaturation process constant as estimated in previous analysis.
The parameters for both model elements were estimated for all three scenarios reliably: the percentage variance accounted for (R2adj) was 99.5 up to 100% (Table 2).
Estimation of all parameters simultaneously only on values for S(t), assuming no
information whatsoever about parameter values of the denaturation process, proved
dicult. That is more a sign that the more measured data are available the better the
estimation procedure can work, rather than that this approach to dynamic behaviour is not working properly.
The data of this theoretical example are also analysed using the temperature sum
in polynomial models, up to a fourth-degree polynomial. In Table 3, the results are
shown of the analysis of the data on substrate during application of the daily scenario, analysed with increasing order of polynomial. The percentage variance
accounted for (R2adj) is acceptable from the third order upward. However, the values
of the coecients are clearly di€erent between successive orders of the polynomial.
As a consequence, the order of the polynomial chosen determines not only the
reliability but the applicability as well.
In Table 4, the results are shown for the classical analysis of substrate as a function
of a fourth-order polynomial in temperature sum. Although each of the scenarios is
Table 3
Results of statistical data analysis with increasing order of polynomial in temperature sums to on one
dynamic scenarioa
Model scenario

Nobs
R2adj
Constant
TS
TS2
TS3
TS4

Substrate daily
Estimate

Estimate

Estimate

Estimate

80
78.8
65.51
ÿ0.0927

80
97.7
88.52
ÿ0.1272
4.9504E-5

80
99.7
96.551
ÿ0.1915
15.209E-5
ÿ0.4287E-7

80
100
99.702
ÿ0.2347
27.371E-4
ÿ1.6107E-7
0.36921E-10

a
Model experiments on the exerted action on a substrate of a denaturating enzyme. See Nomencalture
for de®nition of abbreviations.

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L.M.M. Tijskens, F. Verdenius / Agricultural Systems 66 (2000) 1±15

Table 4
Results of statistical data analysis with fourth order of polynomial in temperature sums on all three
scenariosa
Model scenario

Nobs
R2adj
Constant
TS
TS2
TS3
TS4

Substrate
Heat-cool estimate

Daily estimate

Distribution estimate

80
99.8
95
ÿ0.3354
7.5593E-4
ÿ9.4410E-7
4.5093E-10

80
100
99.702
ÿ0.2347
2.7371E-4
ÿ1.6107E-7
0.36921E-10

80
92.5
120.46
ÿ1.1696
40.462E-4
ÿ18.045E-7
ÿ71.018E-10

a

Model experiments on the exerted action on a substrate of a denaturating enzyme. See Nomenclature
for de®nition of other abbreviations.

described reliably by the model, the parameter values for each of the separate scenarios are very di€erent, sometimes up to one order of magnitude. As a consequence, the results of the analysis of one scenario cannot be used to described or
predict the behaviour of the substrate in another scenario. This limits the generic
applicability of these types of models.
5.4. Consecutive reactions
Consecutive reactions do also occur in nature and are used in fundamentaloriented models (Tijskens et al., 1998b). The simpli®ed mechanism is basically a
combination of two ®rst-order reactions as shown in Eq. (16):
k1

C1 ! C2 ;
k2

C2 ! C3 :

…16†

Based on this mechanism and applying the rules of fundamental kinetics, one arrives
at a set of di€erential equations:
@C1 …t†
ˆ ÿk1  C1 …t†;
@t
@C2 …t†
ˆ k1  C1 …t† ÿ k2  C2 …t†:
@t

…17†

The analytical solution at constant temperatures is:
C1 …t† ˆ C10  eÿk1 t ;
C2 …t† ˆ C20  eÿk2 t ‡ C10  k1 




eÿk1 t ÿ eÿk2 t
:
k2 ÿ k1

…18†

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L.M.M. Tijskens, F. Verdenius / Agricultural Systems 66 (2000) 1±15

Again, data were generated by numerical integration of Eq. (17), using the three
dynamic scenarios, and a value for the input parameters as shown in second column
in Table 5. The behaviour of the intermediate compound C2 at the three dynamic
scenarios is shown in Fig. 3. Again, the factors k1t and k2t can be approximated
by their respective rate sums kS1(t) and kS2(t). The unresolved occurrences of k1 and
k2, not coupled with time, only occur in ratios. They can therefore again be substituted by the respective rate sums. The dynamics of the temperature scenarios can
now be analysed by non-linear regression. In Table 5 the statistical estimates of the
input parameters are shown for each dynamic scenario for each equation of Eq.
(18), analysing both reactions simultaneously and the ®rst reaction separately.
Table 5
Results of statistical data analysis with rate sums to incorporate dynamic scenariosa
Input

Heat-cool
Estimate

S.E.

Daily

Distribution

All scenarios combined

Estimate S.E.

Estimate S.E.

Estimate

R2adj

100
93.4
81
81
Nobs
C10
80
79.9927 0.00763 79.9722
0.0104
k1ref
0.5 0.486461 0.000325 0.472767 0.000149
4.64 10 359.6
15.2
Ea1 10 000 10 027.57
R2adj
Nobs
80
C10
C20
20
k1ref
0.5
Ea1 10 000
k2ref
0.15
Ea2 15 000

100
81
80.407
19.9099
0.45343
9695.6
0.16508
15 856.9

0.198
0.0493
0.00714
74.3
0.00117
87.5

100
81
97.337
3.705
0.44875
8752
0.14684
16 900

0.22
0.124
0.002
273
0.00116
253

99.9
81
79.876
0.486
0.59644 0.00951
10 698.5
81.1
100
81
Many

S.E.

99.7
243
79.87
0.52911
10 094.8

0.311
0.00628
72.9

99.9
243
83.314
19.383
0.5247
10 132.9
0.15454
14 539

0.681
0.302
0.00909
75.1
0.00137
85.4

a
Model experiments on consecutive reactions. S.E., standard error of estimate. See Nomenclature for
de®nition of other abbreviations.

Fig. 3. Intermediate compound C2 of a consecutive reaction for three dynamic scenarios (Eqs. (16)
and (17)).

14

L.M.M. Tijskens, F. Verdenius / Agricultural Systems 66 (2000) 1±15

Again, the estimates for all three scenarios are very satisfactory. For the distribution
scenario, multiple solutions are possible, at least for the statistical estimation package. This con®rms once again the fact that insucient temperature information is
contained in the data of this scenario: roughly only two temperature levels are
present in the scenario. These ®ndings again stress the importance of scenario design
in estimating parameters in a dynamic environment (Versyck et al., 1997a, b, 1998a,
b, 1999; Bernaerts et al., 1998, 2000; Van Impe, 1998; Van Impe et al., 1998).
Since the processes, occurring in the three di€erent scenarios, are fundamentally
the same. It should therefore be possible to analyse the data of these three scenarios
together. The rates sums have then to be calculated over the temperatures of each
scenario separately. In the last column of Table 5 the results of the combined analysis are shown. Now the amount of information on temperature±time combinations
is amply sucient to obtain reliable and unique estimates of the parameters of the
model. This exercise proves the added value of the principle: models based on fundamental relations are indeed reusable in all kind of scenarios.

6. Conclusions
The empirical temperature sum in polynomial modelling can be replaced by a rate
sum, speci®c of each occurring process. The approach proposed allows the development of more fundamental models that can still be applied in dynamic external
factors.
Statistical analysis of the data incorporating all information on the dynamic scenarios is possible on a function level, and no di€erential equations need to be used.
This ensures that the huge package of statistical tools remains available for application in dynamic external conditions.
In dynamic circumstances, multiple solutions for parameter estimation are possible. This makes the understanding of interactions and of e€ects even more important, and a larger emphasis has to be put on the experimental design.
Acknowledgements
This study was partially conducted in the framework of the SPOT-IT2 project.
The ®nancial support of FTK Holland BV, Bleiswijk, The Netherlands, is gratefully
acknowledged as are the valuable comments and suggestions of Prof. Dr. J. Van
Impe, Dr. J. Top and Dr. J.M. Soethoudt.
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