1. The DEB-theory and its applications in Ecotoxicology
Modélisation en écotoxicologie
Modelleren in de ecotoxicologie
Modeling in ecotoxicology
The DEB-theory and its applications in Ecotoxicology
Jacques J.M. Bedaux
Dept Theoretical Biology
Vrije Universiteit Amsterdam

2. Overview DEB-theory DEBtox modeling frame work Some examples
Extensions to standard DEBtox
New developments
(demo of DEBtox)

3. DEB theory
Dynamic Energy Budget theory
History:
It started as a statistical methodology for the study of toxic effects on Dapnnia reproduction.
DEB theory presents simple mechanistic rules that describe the uptake and use of energy and nutrients (substrates, food, light) and the consequences for physiological organisation throughout an organism's life cycle
History of the DEB theory: as Bas Kooijman writes in the preface of his first book
it started in 1978 as a statistical methodology for the study of toxic effects on daphnid reproduction!
Large daphnids tend to have bigger litters (i.e. they produce more offspring) than small ones, so modelling reproduction is impossible without looking at growth, feeding etc.
So it started as a model for Daphnia, but evolved into a non-species-specific theory.
It is impossible too say much about DEB-theory in half an hour.
I’ll give some information about it, and

4. DEB model (simple form)
food reserve structure
maturation maint. somatic maint.
reproduction k
1-k
DEB model for a reproducing heterotroph (such as many animals);
Structural body mass and reserves are the state variables of the individual;
The kappa-rule: a fixed fraction of the energy, used from the reserves, is spent on somatic maintenance plus growth.
Maintenance is proportional to structural body volume.
Feeding rate is proportional to the surface area of the body.
Under starvation conditions, individuals always give priority to somatic maintenance.
Feeding starts at the transition from embryo to juvenile, and the transition to adult initiates reproduction

5. The DEB model The model is tested on a large amount of empirical data.
Daphids dataset:
Ingestion rate : proportional with f.L^2
Oxygen-consumption rate: proportional with vL^2 + km.L^3
Growth: Von Bertalanffy

6. framework of toxicological model
external concentration of chemical compound
Standard way of toxicological modeling:
The toxicological endpoint, like reproduction, is taken as a function of environmental concentration, without incorporating time of exposure. This way of static modeling is easy to apply, but not very realistic and without biological motivation.
toxicological endpoint
(growth, reproduction, survival)

7. static toxicological model
Log-logistic model
Standard way of toxicological modeling:
The toxicological endpoint, like reproduction, is taken as a function of environmental concentration, without incorporating time of exposure. This way of static modeling is easy to apply, but not very realistic and without biological motivation.

8. dynamic toxicological model
external concentration
toxicokinetics
internal concentration
effects (toxicodynamics)
physiological parameter
blanc physiology
toxicological endpoint

9. framework standard DEBtox model
external concentration
toxicokinetics
first-order
internal concentration
effects
linear
physiological parameter
blanc physiology
DEB model
toxicological endpoint

10. Example: acute test, survival
external concentration
C’(t) = kuc – keC(t)
internal concentration
Hazard rate proportional to internal concentration minus threshold
h(t)  (C(t) – threshold)+
The most simple example is an acute survival test. Short term test, so we can neglect growth of the animals.
We relate the fraction of survivors (the survivor function) directly to the internal concentration by stating that the hazard rate is proportional tot the internal concentration, as far as it exceeds an internal threshold.
We believe that an animal can handle some amount of toxicants without any damage.
By the way, this threshold might be equal to zero for some species/toxicant combination.
In this example DEB theory is not really present, only the DEB philosophy!
S(t) = exp(-0th(u)du)
fraction of survivors

11. internal concentration
c > NEC
internal threshold
c = NEC
c < NEC
time
hazard rate
survivor function
time
time

12. time-dose response surface
dose response curve
time-dose response surface
time
concentration
concentration
4 parameters
NEC
elimination rate
killing rate
background mortality

13. Example: chronic test, reproduction
food
assimilation
reserve
The most complex test we have worked out is the chronic test on effects on reproduction.
Here the DEB model is indispensable.
DEB theory describes reproduction rate as a function of length (by constant feeding). We assume that the structure of the DEB model is not affected by a toxicant, only some parameters values are changed. For instance
growth
maintenance
reproduction
structure
work
eggs

14. Effects on reproduction costs
external concentration
First order, dilution by growth
internal concentration
Energy costs per egg proportional to internal concentration minus threshold
Here we look at one of the five modes of action: effects on reproduction costs.
DEB is involved in all levels. It leads to a system of three coupled differential equations, describing length, concentration en total reproductive output. In standard experiments we only have data on reproductive output. Four parameters have to be estimated.
energy costs per egg
reproduction rate

15. Chronic tests: reproduction
DEBtox specifies length, internal concentration and reproduction rate
data only on reproductive output!
4 parameters to be estimated:
- NEC, tolerance concentration,
- elim. rate, max. reproduction rate
numerically heavy

16. Free software: DEBtox Four types of toxicity tests - Survival
- (Fish) Body length growth
- (Daphnia) Reproduction
- (Algae) Population growth
Parameter estimation, confidence intervals, likelihood profiles, statistical tests, graphical output
Windows, Unix
- User friendly (Windows)
- For experts (Unix)
downloadable on

17. Extensions to standard DEBtox
From effects on individuals to effects on population level
Time-inhomogeneous concentrations (Alexandre Péry)
More-samples analysis

18. From individual to population
Euler-Lotka equation
mc/m0 c
EC10

19. References References on DEB:
Kooijman, S.A.L.M. (2000) Dynamic Energy and Mass Budgets in Biological Systems (2nd ed) Cambrige UP.
Kooijman, S.A.L.M. (2000) Quantitative aspects of metabolic organization: a discussion of concepts, Phil Trans R Soc LondB 356
Nisbet, R.M. et al (2000) From molecules tot ecosystems through dynamic energy budget models, Journal of Animal Ecology 69
References on DEBtox:
Kooijman, S.A.L.M. and Bedaux, J.J.M. (1996) The analysis of aquatic toxicity data, VU UP.

20. Related projects Biodegradation (DEBdeg) Tumor incidence (DEBtum)
Food-webs