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\title{Ecotoxicological applications of Dynamic Energy Budget theory
  \footnote{subm \emph{J. Devillers (ed) Ecotoxicological
      Modelling, Springer}}}
% max length 15625 words

\author{S.A.L.M. Kooijman\footnote{corresponding author: bas@bio.vu.nl. 
  File-name: kinetics.tex}, J. Baas, D. Bontje, M. Broerse,\\
  C. A. M. van Gestel, and T. Jager\\[5mm]
  {\footnotesize Faculty Earth \& Life Sciences} \\
  {\footnotesize Vrije Universiteit, de Boelelaan 1085, 1081 HV
    Amsterdam, NL}}

\date{}

\maketitle

\newpage

    {\abstract{The Dynamic Energy Budget (DEB) theory for metabolic
    organisation specifies quantitatively the processes of uptake of
    substrate by organisms and its use for the purpose of maintenance,
    growth, maturation and reproduction. It applies to all organisms.
    Animals are special because they typically feed on other
    organisms. This couples the uptake of the different required
    substrates, and their energetics can, therefore, be captured
    realistically with a single reserve and a single structure
    compartment in biomass. Effects of chemical compounds (e.g.\
    toxicants) are included by linking parameter values to internal
    concentrations. This involves a toxico-kinetic module that is
    linked to the DEB, in terms of uptake, elimination and (metabolic)
    transformation of the compounds. The core of the kinetic module is
    the simple one-compartment model, but extensions and modifications
    are required to link it to DEBs. We discuss how these extensions
    relate to each other and how they can be organised in a coherent
    framework that deals with effects of compounds with varying
    concentrations and with mixtures of chemicals. For the
    one-compartment model and its extensions, as well as for the
    standard DEB model for individual organisms, theory is available
    for the co-variation of parameter values among different
    applications, which facilitates model applications and
    extrapolations.}}

\noindent
\emph{Keywords:} Dynamic Energy Budgets, effects on processes,
kinetics, metabolism, transformation

\newpage

\section{Introduction}

The societal interest in ecotoxicology is in the effects of chemical
compounds on organisms, especially at the population and ecosystem
level. Sometimes these effects are intentional, but more typically
they concern adverse side-effects of other (industrial)
activities. The scientific interest in effects of chemical compounds
is in perturbating the metabolic system, which can reveal its
organisation.  This approach supplements ideas originating from
molecular biology, but now applied at the individual level; the sheer
complexity of biochemical organisation hampers reliable predictions of
the performance of individuals.  Understanding the metabolic
organisation from basic physical and chemical principles is the target
of Dynamic Energy Budget (DEB) theory \cite{Kooy2000,Kooy2001}. In
reverse, this theory can be used to quantify effects of chemical
compounds, i.e.\ \emph{changes} of the metabolic performance of
individuals. This chapter describes how DEB theory quantifies toxicity
as a process.

The effects can usually be linked to the concentration of compounds
inside the organism or inside certain tissues or organs of an
organism. This makes that toxicokinetics is basic to effect studies.
The physiological state of an organism, such as its size, its lipid
content, and the importance of the various uptake and elimination
routes (feeding, reproduction, excretion) interacts actively with
toxicokinetics, so a more elaborate analysis of toxicokinetics should
be linked to the metabolic organisation of the organism \cite{Kooy97}.
In the section on toxicokinetics, we start with familiar classic
models that hardly include the physiology of the organism, and
stepwise include modules that do make this link in terms of logical
extensions of the classic models.

Three ranges of concentrations of any compound in an organism can be
delineated: too-little, enough and too-much. The definition of the
enough-range is that variations of concentrations within this range do
not translate into variations of the physiological performance of the
individual. Some of the ranges can have size zero; such as the
too-little-range for cadmium. Effects are quantified in the context of
DEB theory as changes of (metabolic) parameters as linked to changes
in internal concentrations. These parameters can be the hazard rate
(for lethal effects), the specific food uptake rate, the specific
maintenance costs, etc. Changes in a single parameter can have many
physiological consequences for the individual. DEB theory is used not
only to specify the possible modes of action of a chemical compound,
but also how the various physiological processes interact. An increase
in the maintenance costs by some compound, for instance, reduces
growth, and since food uptake is linked to body size, it indirectly
reduces food uptake, and so affects reproduction (and
development). The existence of the ranges too-little, enough and
too-much of concentrations of any compound directly follows from a
consistency argument, where no classification of compounds is accepted
(e.g.\ toxic and non-toxic compounds), and many compounds (namely
those that make up reserve) exist that do change in concentration in
the organism, without affecting parameter values. An important
consequence is the existence of an internal No Effect Concentration (NEC),
as will be discussed below.

Effects at the population level are evaluated from those at the
individual level, by considering populations as a set of interacting
individuals [4-8]\nocite{KooyAnde2004,KooyGras2007,KooiKelp2003,KooyKooi99,NisbMull97}.
Although DEB-based population dynamics can be complex, particular
aspects of population performance, such as the population growth rate
at constant environmental conditions, are not very complex. Moreover
particular simplifying approximations are possible.  The focus of
toxicology is typically at time scales that are short relative to the
life span of the individual. That of ecotoxicology, however, are much
longer, involving the whole life history of organisms; effects on
feeding, growth, reproduction and survival are essential and typically
outside the scope of toxicology. This has strong implications for the
best design of models. While pharmacokinetics models frequently have
many variables and parameters, such complex models are of little use
for applications at the population level, where a strong need is felt
for relatively simple models, but then applied to many species and in
complex situations. DEB theory is especially designed for this task.

Chemical transformations are basic to metabolism, and transformations
of toxicants are no exception. Compounds that dissociate in water
should be considered as a mixture of ionic and molecular forms, and
the pH affects that mixture. This makes that the effect of a single
pure compound is of rather academic interest; we have to think in
terms of the dynamic mixtures of compounds. Because of its strong
links with chemical and physical principles, DEB theory has
straightforward ways to deal with effects of mixtures. Some of this
theory rests on the covariation of parameter values across species of
organism and across chemical compounds. Theory on this covariation is
implied by DEB theory, and is one of its most powerful aspects.

We first introduce some notions of DEB theory, and then discuss
toxicokinetics and effects in the context of DEB theory.

\section{The standard DEB model in a nutshell}

\begin{figure}[tb]
\setlength{\unitlength}{1cm}
\begin{picture}(8,4)(-1,-.3)\footnotesize
  \put(0,3){\oval(1.5,0.5)} \put(0,3){\makebox(0,0){food}}
  \put(0.75,3){\vector(1,0){4.5}}
  \put(6,3){\oval(1.5,0.5)} \put(6,3){\makebox(0,0){faeces}}
  \put(2.75,2.75){\oval(0.5,0,5)[tr]} \put(3,2.75){\vector(0,-1){0.3}}
  \put(3.75,2.1){\makebox(0,0){\framebox(1.5,0.7)}}
  \put(3,2.1){\makebox(0,0)
    {$\begin{array}{cc} \mbox{reserve} \\[-1mm] M_E \end{array}$}}
  \put(3,1.75){\line(0,-1){0.15}} \put(3,1.5){\circle{.2}}
% \put(2,3.25){\makebox(0,0){$X$}} \put(4,3.25){\makebox(0,0){$P$}}
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  \put(2.9,1.4){\vector(-3,-1){3}}
  \put(0,0){\oval(1.5,0.75)} \put(0,0){\makebox(0,0)
    {$\begin{array}{cc} \mbox{somatic} \\[-1mm] \mbox{work} \end{array}$}}
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  \put(2,1.1){\circle{.1}}
  \put(4.75,0){\makebox(0,0){\framebox(1.5,0.75)}}
  \put(4,0){\makebox(0,0) 
  {$\begin{array}{cc} \mbox{structure} \\[-1mm] M_V \end{array}$}}
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  \put(6,0){\makebox(0,0)
  {$\begin{array}{cc} \mbox{maturity} \\[-1mm] M_H \end{array}$}}
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    \mbox{work} \end{array}$}}
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  \put(6.2,1.65){\vector(-1,1){0}}
\end{picture}\hspace{1cm}\footnotesize
\parbox[b]{6cm}{Fluxes of food and reserve \\[1mm]
  \begin{tabular}{ccl}\hline
  1 & feeding\\
  2 & assimilation\\
  3 & mobilisation\\
  4 & somatic maintenance\\
  5 & maturity maintenance\\
  6 & reproduction\\
  \hline
  \end{tabular}\\}
\caption[The standard DEB model with
  fluxes (moles per time) and pools (moles). Assimilation is zero
  during the embryo stage and becomes positive at the transition to
  the juvenile stage (birth) if food is available. Age is zero at the
  start of the embryo stage. Reproduction is zero during the juvenile
  stage and becomes positive at the transition to the adult stage
  (puberty), when further investment into maturation is ceased.]
  {\label{fig:flow} \protect\small}
\end{figure}

The standard DEB model concerns an isomorph. i.e.\ an organism that
does not change in shape during growth, that feeds on a single food
source (of constant chemical composition), and has a single reserve a
single structure and three life stages: embryo (which does not feed),
juvenile (which does not allocate to reproduction) and adult (which
allocates to reproduction, but not to maturation). This is in some
respects the simplest model in the context of DEB theory, which is
thought to be appropriate for most animals. Food is converted to
reserve, and reserve to structure. Reserve does not require
maintenance, but structure does, mainly to fuel its turnover, see
Figure \ref{fig:flow}. Reserve can have active metabolic functions and
serves the role of representing metabolic memory. Reserve and
structure do not change in chemical composition (strong
homeostasis). At constant food availability, reserve and structure
increase in harmony, i.e.\ the ratio of their amounts, the reserve
density, remains constant (weak homeostasis).

The shape (and so the change of shape) is important because food
uptake is proportional to surface area, and maintenance mostly to
(structural) volume. The handling time of food (including digestion
and metabolic processing) is proportional to the mass of food
``particles'', during which food acquisition is ceased. The
mobilisation rate of reserve to fuel the metabolic needs follows from
the weak and strong homeostasis assumptions \cite{SousDomi2008}; a
mechanism is presented by Kooijman and Troost \cite{KooyTroo2007}.
Allocation to growth and somatic maintenance (so to the soma)
comprises a fixed fraction of mobilised reserve, the remaining
fraction is allocated to maturation (or reproduction) and maturity
maintenance.

The transition from the embryo to the juvenile stage (i.e.\ birth)
occurs by initiating assimilation when the maturity exceeds a
threshold value, and from the juvenile to the adult stage (i.e.\
puberty) by initiating allocation to reproduction and ceasing of
allocation to maturation at another threshold value for
maturity. Reserve that is allocated to reproduction is first collected
in a buffer, that is subjected to buffer handling rules (such as
spawning once per season, or convert the buffer content into an egg as
soon as there is enough).

Biomass consists of reserve and structure, and can, therefore, change
in chemical composition (e.g.\ lipid content) in response to the
nutritional condition; maturity has the status of information, not
that of mass or energy. Apart from some minor details, the presented
set of simple rules fully specify the dynamics of the individual,
including all mass and energy fluxes, such as the uptake of dioxygen,
the production of carbon dioxide, nitrogen waste and heat. It takes
some time to see exactly how; Sousa \emph{et al.} \cite{SousDomi2008}
gives a nice evaluation.  Aging is considered to result from a side
effect of Reactive Oxygen Species (ROS), and is so linked to the
uptake of dioxygen \cite{LeeuKelp2002}. A high food uptake results in
a large amount of reserve, so a high use of reserve, a high uptake of
dioxygen, an acceleration of aging and a reduction of life span. The
induction of tumours is also linked to the occurrence of ROS, and
other reactive molecules, such as mutagenic compounds. This gives a
natural link between aging and tumour induction.

The standard DEB model has been extended into many directions for the
various purposes. The allocation rule to the soma, for instance, can
be refined to allocation to various body parts (e.g.\ organs), where
growth of each body part is proportional to the allocated reserve flux
minus what is required for maintenance of that part. Rather than using
fixed fractions of the mobilised reserve, the fractions can be linked
to the relative workload of the body part. This allows a dynamic
adaptation of the body parts in interaction with their use. Tumours
can be considered as body parts, and the ``workload'' of the tumour is
the consumption of maintenance. This formulation produced realistic
predictions of the effects of caloric restriction on tumour growth,
and of the growth of tumours in young versus old hosts
\cite{LeeuZonn2003}. This approach can be extended to various types of
tumours, where tumour growth is not linked to that of the whole body,
but that of a particular body part. Many tumours result from
destruction of local cell-to-cell communication, rather than from
genotoxic effects, but these different routes have similar dynamics.

Other types of extensions of the standard DEB model concern the
inclusion of variations in chemical composition of food (with
consequences for the transformation of food into reserve) and
size-dependent selection of different food items. For example many
herbivores are carnivores when young. Animals are special because they
feed on other organisms. Most other organisms take the food-compounds
(energy source, carbon source, nitrate, phosphate) that they need
independently from the environment, which necessitates the inclusion
of more than one reserve; see Kooijman and Troost \cite{KooyTroo2007}
for the evolutionary perspectives. Most micro-organisms grow and
divide, and don't have the three life stages delineated by the
standard model. This makes that their change in shape hardly matters
and that surface areas can be taken proportional to volumes, which
simplifies matters considerably. The partial differential equations
that are required to described the physiologically structured
population dynamics of isomorphs then collapse to a small set of
ordinary differential equations. Plants, on the other hand, require at
least two types of structure (roots and shoots) and have a complex
adaptive morphology (i.e.\ surface area-volume relationships); their
budgets are most complex to quantify.

\section{Family of toxicokinetic models}

Originally (before the 1950's) the focus of toxicology, i.e.\ the
field that gave rise to ecotoxicology, was on medical applications of
compounds in a pharmacological context. Subjects where given a
particular dose, and the interest is in the redistribution inside the
body, and in transformation and elimination. The aim is to reach the
target organ and to achieve a particular effect that restores the
health or well-being of the subject. A closely related interest that
developed simultaneously was health protection (disinfection and food
protection products), with the purpose of killing certain species of
pathogenic organism (especially microorganisms), or to reduce their
impact.

After the 1950's ecotoxicology began to flourish and gradually became
more independent of toxicology, where the initial focus was in
positive and (later) negative effects of biocides. This came with
extensions of the interest in the various uptake and elimination
routes that are of ecological relevance, and the environmental
physical-chemistry of transport and transformation. The aim is to kill
particular species of pest organism locally (insects, weeds), and to
avoid effects on other, non-target, species (crop and beneficial
species). After the 1970's the interest further generalised to an
environmental concern of avoiding effects of pollutants on organisms,
with an increasing attention for (bio)degradation of compounds that
are released into the environment, coupled to human activity
\cite{BracBakk2005}.

This historic development and branching of the interest in
toxico-kinetics came with a narrowing of the focus on a particular
aspect of toxico-kinetics in the various applications that, we feel,
is counter-productive from a scientific point of view. The purpose of
this section is an attempt to restore the coherence in the field, by
emphasising the general eco-physiological context and the relationship
of the various models with the core model for toxico-kinetics: the
one-compartment model. The more subtle models account for the
interaction with the metabolism of the organism, which involves its
metabolic organisation.

We here focus on the logical coherence of toxico-kinetics,
bio-availability and metabolism, including effects (= changes in
metabolism). It is not meant to be a review. For recent reviews on
toxico-kinetic models, see o.a.\ Barber \cite{Barb2003} and Mackay and
Fraser \cite{MackFras2000}.  See Table \ref{tab:symbols} for a list of
frequently used symbols.

\begin{table}[tb]
\caption[List of frequently used symbols, with units and interpretation.
  The unit C-mol represents the number of C-atoms in a organism as
  multiple of the number of Avogadro.]
  {\label{tab:symbols}\protect\small}

\centerline{\small
\begin{tabular}{lll}\hline
  symbol & units & interpretation \\ \hline
  $t$ & d & time\\
  $c$ & M & concentration of compound in the environment\\
  $n_i$ & mol\,m$^{-1}$ & density of compound\\
  $Q$ & mol\,C-mol$^{-1}$ & concentration of compound in an organism\\
  $r$ & d$^{-1}$ & specific growth rate of structure\\
  $k_e$ & d$^{-1}$ & elimination rate\\
  $k_{01}$, $k_{10}$ & d$^{-1}$ & exchange rates between compartments\\
  $P_{0d}$ & mol\,C-mol$^{-1}$\,M$^{-1}$ & BioConcentration Factor (BCF)\\
  $v_{ij}$, $v_e$ & m\,d$^{-1}$ & velocity, elimination -\\
  $d_i$ & m$^2$\,d$^{-1}$ & diffusivity\\
  $m_E$, $m_{ER}$ & C-mol\,C-mol$^{-1}$ & reserve density, reproduction buffer 
    density\\
  \hline
\end{tabular}}
\end{table}

\subsection{One-compartment model}

\begin{figure}
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\begin{picture}(5.5,3)\small
  \put(1, .75){\vector(1,0){2}}
  \put(4, 1){\makebox(0,0){\framebox(1.5,1.5){\large $Q$}}} 
  \put(0.5, 1){\makebox(0,0){\large $c$}} 
  \put(3, 1.25){\vector(-1,0){2}}
  \put(2, .2){\makebox(0,0){$k_e P_{0d}$}} 
  \put(2, 1.8){\makebox(0,0){$k_e$}} 
\end{picture}
\begin{picture}(8,3)\small
  \put(1, .75){\vector(1,0){2}}
  \put(4, 1){\makebox(0,0){\framebox(1.5,1.5){\large $Q_0$}}} 
  \put(0.5, 1){\makebox(0,0){\large $c$}} 
  \put(3, 1.25){\vector(-1,0){2}}
  \put(2, .2){\makebox(0,0){$k_e P_{0d}$}} 
  \put(2, 1.8){\makebox(0,0){$k_e$}} 
  \put(5, .75){\vector(1,0){2}}
  \put(8, 1){\makebox(0,0){\framebox(1.5,1.5){\large $Q_1$}}} 
  \put(7, 1.25){\vector(-1,0){2}}
  \put(6, .2){\makebox(0,0){$k_{01}$}} 
  \put(6, 1.8){\makebox(0,0){$k_{10}$}} 
\end{picture}
\caption[The scheme of the one- and
  two-compartment models. The factor $P_{0d}$ converts an external
  concentration into an internal one; all rates labelled $k$ have
  dimension `per time'. In the two-compartment model $P_{0d}$ does not
  have the interpretation of the bioconcentration factor.]
  {\label{fig:1-comp}\protect\small}
\end{figure}

The core model in toxico-kinetics is the one-compartment model, see
Fig. \ref{fig:1-comp}. It states that the uptake rate is proportional
to the environmental concentration $c$, and the elimination rate is
proportional to the internal concentration $Q$:

\begin{eqnarray}
  \label{eqn:1-comp}
  \frac{d} {dt} Q &=& k_e \left( P_{0d} c(t) - Q \right)
\\
  Q(t) &=& Q(0) \exp(- t k_e) +  
           k_e P_{0d} \int_0^t c(s) \exp( (s - t) k_e) \, ds
\\
       &=& Q(0) \exp(- t k_e) +  P_{0d} c \left( 1 - \exp( - t k_e) \right)
           \quad \mbox{for $c$ is constant}
\end{eqnarray}
where $P_{0d} = Q(\infty)/ c$ is the BioConcentration Factor (BCF) and
$k_e$ the elimination rate. The product $k_e P_{0d}$ is known as the
uptake rate, and is frequently indicated with $k_u$, which is
misleading because its units are d$^{-1}$\,m$^3$\,C-mol$^{-1}$. Even
if we would work with kg rather than C-mol, and the specific density
of the organism equals 1 kg\,dm$^{-3}$, the BCF is not dimensionless
\cite{Kooy2000}. It is typically more convenient to work with
molalities in soils (mol\,kg$^{-1}$), and with molarities
(mol\,l$^{-1}$) in water. Molalities give the uptake rate the units
d$^{-1}$\,g\,C-mol$^{-1}$. Many workers use gram rather than mole to
quantify the compound, but this choice is less practical to compare
the toxicity of different compounds. The elimination rate $k_e$ has
dimension `per time' and is indepenendent of how the compound is
quantified; contrary to the uptake rate, the elimination rate can be
extracted from effect data and determines how fast effects build up in
time, relative to the long-term effect level.

The concentrations $c$ and $Q$ must obviously exist, meaning that the
environment and the organism are taken to be homogeneous. This
condition can be relaxed without making the model more complex by
allowing a spatial structure (such as organs), and an exchange between
the parts that is fast relative to the exchange between the organism
and the environment. Other implicit assumptions of the one-compartment
model are that the organism does not change in size or in chemical
composition, so changes in food availability must be negligible. The
(bio)availability of the compound remains constant. So transformations
can be excluded, and the environment is large relative to the organism
and well-mixed. Sometimes, e.g.\ in the case of a large fish in a
small aquarium, this is not true and the dynamics of the concentration
in the organism and the environment should be considered
simultaneously: the 1-1 compartment model \cite{KooyJage2004}. These
restrictions will be removed below.

Since rates generally depend on temperature, and temperature typically
changes in time, the elimination rate $k_e$ can change in time as well
\cite{JansBerg91}. In the sequel we will discuss rates of metabolism,
and like all rates, these also depend on temperature, frequently
according to the Arrhenius relationship \cite{Kooy2000}.

\subsection{Multi-compartment models}

If transport inside the organism is not fast, relative to the exchange
with the environment, multiple-compartment models should be considered
\cite{Godf83,Jacq72}. If exchange with the environment is only via
compartment number 0, the change in the concentrations in the nested
compartments number 0 and 1 is

\begin{equation}
  \label{eqn:n-comp}
  \frac{d} {dt} Q_0 = k_e \left( P_{0d} c - Q_0 \right) + 
     k_{10} Q_1 - k_{01} Q_0; \quad
  \frac{d} {dt} Q_1 = k_{01} Q_0  - k_{10} Q_1
\end{equation}
where $k_{01}$ and $k_{10}$ are the exchange rates between the
compartments, see Fig. \ref{fig:1-comp}. The partition coefficient
between the compartments equals $P_{10} \equiv Q_1(\infty)/
Q_0(\infty) = k_{01}/ k_{10}$, while $P_{0d} \equiv Q_0(\infty)/ c$
remains unchanged. In many cases whole body measurements are used. If
$M_0$ and $M_1$ denote the masses of compartment 0 and 1, the whole
body partition coefficient with the environment amounts to $P_{+d} =
(M_0 + M_1 P_{10}) P_{0d}/ M_+$, with $M_+ = M_0 + M_1$ is the whole
body mass. This rather complex behaviour of the whole body partition
coefficient can be a source of problems in fitting models to data. In
many practical cases it is not possible to identify the compartments
and to measure the concentrations in these compartments directly. The
use of multi-compartment models cannot be recommended in such cases.

This extension still classifies as a transport model, so in a clean
environment ($c = 0$), the organism will loose all its load
($Q_0(\infty) = Q_1(\infty) = 0$). Quite a few data sets on the
kinetics of (``heavy'') metals in organisms show that once loaded, an
organism never fully looses its load [20-22]\nocite{SpurHopk99,ShepEven97,VijvVink2005}. Such behaviour cannot be
captured by multi-compartment models because this involves an
extension to transformation of compounds (technically speaking, sequestered
compounds belong to a different chemical species).

If $k_{01}, k_{10} \gg k_e$, we can assume that $Q_1(t) \simeq P_{10}
\, Q_0(t)$, and the kinetics Eq~\ref{eqn:n-comp} reduces to
Eq~\ref{eqn:1-comp}, with $Q$ replaced by $Q_0$. This situation is
called time-scale separation.

The interpretation of the compartments can be a special tissue or
organ, or, more abstract, reserve (compartment 1) and structure
(compartment 0). The latter makes sense in the context of DEB theory,
where both compartments are assumed to have a constant, but different,
chemical composition, while reserve is relatively rich in lipids in
many animal taxa. While Eq~\ref{eqn:n-comp} assumes that the size of
all compartments does not change, we will relax on this below, when we
allow more interactions with metabolism and energetics.

We can obviously include more compartments, and more complex
interactions with the environment, but the number of parameters
rapidly increases this way. In practice multi-compartments are used
if the one-compartment model fits data badly. The introduction of more
parameters generally improves the fit, but not necessarily for the
right reasons. As a rule of thumb it is only advisable to use more
compartment models if data on the concentrations inside the
compartments are available. If lack of fit of the one-compartment
model is the only motivation, alternatives should be considered that
are discussed below.

\subsection{Film models}

Film models are conceptually related to multi-compartment models
because both are extensions of the one-compartment model that include
more detail in transport (so in physical factors), though in different
but complementary ways. Film models are especially popular in
environmental chemistry for following the transport of compounds from
one environmental compartment (such as water) to another (such as
air). Both compartments are assumed to be well-mixed, except for a
narrow film at the interface of both compartments, where transport is
by diffusion.

To follow the dynamics for the densities of the compound $n$ (mol/m),
we need to define a spatial axis perpendicular to the interface and
choose the origin at the boundary between the bulk and the film (on
each side of the interface). Let $L_i$ be the depth of the film, $d_i$
the diffusivity of the compound in the film, and $v_{ij}$ the exchange
velocity of the compound between the two media. As discussed in
Kooijman \emph{et al.} \cite{KooyJage2004}, the dynamics of the
densities is given by partial differential equations (pde's) for
medium $i = 1 - j$ and $j = 0$ or $1$

\begin{equation}\label{eqn:part}
  0 = \frac{\partial} {\partial t} n_i(L,t) - 
      d_i \frac{\partial^2} {\partial L^2} n_i(L,t) 
      \quad \mbox{for } L \in (0,L_i)
\end{equation}
with boundary conditions at $L = 0$ (i.e.\ the boundary between the
film and the well-mixed medium) for $v_i = d_i/ L_i$

\begin{equation}\label{eqn:boundaryi}
  0 = \frac{\partial} {\partial t} n_i(0,t) -
      v_i \frac{\partial} {\partial L} n_i(0,t)
\end{equation}
and boundary conditions at $L = L_i$ (i.e.\ the interface between the
media where the two films meet)

\begin{equation}\label{eqn:boundaryij}
  0 = v_{ji} n_j(L_j, t) - v_{ij} n_i(L_i, t) + 
      d_{i} \frac{\partial} {\partial L} n_i (L_i,t)
\end{equation}
The boundary condition at $L = 0$, and the diffusion process in the
film is rather standard, but we believe that the boundary condition at
$L = L_i$ is presented for the first time in Kooijman \emph{et al.}
\cite{KooyJage2004}. Users of the popular film models typically skip the
formulation of the pde and directly focus at steady state situations;
they typically use the concentration jump across the interface that
belongs to the situation when there is no net transport across the
interface. As long as there is transport, however, the concentration
jump differs from this equilibrium value.

The depth of the films is typically assumed to be small and the
transport in the films in steady state, which makes that the density
profiles in the films are linear. This leads to the 1-1-compartment
kinetics for the bulk densities 

\begin{equation}
  \frac{d} {dt} n_i(0) = k_{ij} (P_{ij} n_j(0) - n_i(0)); \quad
  k_{ij} = \frac{v_i/ \mathcal{L}_i} {1 + P_{ij} v_i/ v_j - v_i/ v_{ij}}
\end{equation}
where $\mathcal{L}_i$ is the depth of the medium. This approximation
only applies if $v_i v_j < v_{ij} v_j + v_{ji} v_i$ and the transport
in the film is rate limiting. The 1-1-compartment kinetics also
results, however, if the film depths reduce to zero and if the
diffusivities are high. The rate from $i$ to $j$ then reduces to $k_{ij} =
v_i {\cal L}_i^{-1} (1 - v_i/ v_{ij})^{-1}$. In these two situations,
transport in the film is no longer rate limiting.

The applicability of film models to toxico-kinetics in organisms is
still an open question. It can be argued that a stagnant water film
sticks to aquatic or soil organisms (and air to a terrestrial
organism), and that the skin (or cuticula) is not well served by the
internal redistribution system (blood) of the organism. If
toxico-kinetics is fully limited by transport in the film, and if it
is not limited by the film, one-compartment kinetics results; only in
the intermediary situation we can expect some deviations. Yet the
discussion is not completely academic, since these details matter for
how the elimination rate depends on the partition coefficient
\cite{KooyJage2004,KooyBaas2007}.

\subsection{Uptake and elimination routes}

We now consider extensions of the one-compartment model due to
biological factors by accounting for various uptake and elimination
routes. These routes depend on the type of organism, its habitat and
properties of the compound. Animals that live in (wet) soil are in
intense contact with the water film around soil particles, and their
situation has similarities with that of aquatic animals. Direct
transport through the skin can be important, which involves the
surface area of an organism. Some parts of the skin are more
permeable, especially that used by the respiratory system for dioxygen
uptake and carbon dioxide excretion. The uptake rate might be linked
to the respiratory rate, which depends on the energetics of the
organism. Generally, the respiration rate scales with a weighted sum
of surface area and volume, but the proportionality constants depend
on the nutritional conditions of the organism \cite{Kooy2000}. For
terrestrial animals, uptake via the lungs from air and via skin
contact with the soil must be considered. Sometimes uptake is via
drinking; the DEB theory quantifies drinking via the water balance for
the individual and involves a.o. metabolism and transpiration. The
details can be found in Kooijman \cite{Kooy2000}.

A second important uptake route is via food and the gut epithelium.
The feeding rate depends on food availability, food quality, and the
surface area of the organism \cite{Kooy2000}.

The elimination can follow the same routes as uptake, but there are
several additional routes to consider, namely via products of
organisms. The first possibility is the route that excreted nitrogen
waste follows (urination). Reproductive products (mostly eggs and
sperm) can also be an important elimination route. Moulting (e.g.
ecdysozoans, including the rejection of gut epithelium, e.g.\
collembolans) or the production of mucus (e.g.\ lophotrochozoans) or
milk (e.g.\ female mammals) are other possible excretion routes. The
DEB theory quantifies reproductive and other products as functions of
the amounts of reserve and structure of the individual. In the
standard DEB model they work out to be cubic polynomials in body
length, but the coefficients depend on the nutritional conditions
(amount of reserves per structure) \cite{Kooy2000}.

\subsection{Changes in body size and composition}

The body size of an organism matters in the context of toxico-kinetics
for several reasons \cite{KooyHare90}. As exchange is via surface
area, and is proportional to concentration, surface area-volume
interactions are involved. This problem also applies to compartment
and film models, but gets a new dimension if we consider changes in
body size, which are linked to the nutritional condition of organisms
(lipid content), and so to (changes in) body composition. Small
changes in size can have a substantial effect on the shape of
accumulation curves.

If an organism does not change in shape during growth (so it remains
isomorphic), surface area is proportional to volume$^{2/3}$, or to
squared length. Moreover, dilution by growth should be taken into
account, even at low growth rates. This modifies Eq~\ref{eqn:1-comp}
to

\begin{equation}
  \label{eqn:1-comp-L}
  \frac{d} {dt} Q = \left( P_{0d} c(t) - Q \right) v_e/ L 
      - Q r \quad \mbox{with} \quad r = \frac{d} {dt} \ln L^3
\end{equation}
where $L$ is the length, and $v_e = L_m k_e$ is the elimination
velocity for maximum length $L_m$. The last term, $Q r$, represents
the dilution by growth. If it equals zero, we can replace $v_e/ L$ by
the constant elimination rate $k_e^\prime$, but its meaning still matters if
we compare the kinetics in two organisms of different size. DEB
theory specifies how the change in (cubed) length depends on the
amount of reserve and structure of the organism, and how the change in
reserve depends on these state variables and food availability. Food
intake and maintenance play an important role in growth and together
they control the maximum size an organism can reach, since food intake
is proportional to a surface area and maintenance to structural
volume. Wallace had this insight in 1865 already \cite{Finc90}.

The DEB theory allows for particular changes in body composition,
because reserve and structure can change in relative amounts and both
have a constant composition. Food (substrate) is first transformed
into reserve, and reserve is used for metabolic purposes, such as
somatic and maturity maintenance, growth, maturation and reproduction.
The change in reserve density for metabolic use is proportional to the
reserve density per length, which makes that high growth rates come
with high reserve densities, i.e. the ratio of the amounts of reserve
and structure.

Reserves are in many animal taxa relatively enriched in lipids, which
might have a strong influence on the kinetics of hydrophobic
compounds. The body burden of eel in a ditch that is polluted with
mercury or PCB might greatly exceed that of other fish partly because
eel is relative rich in lipids. This illustrates the importance of
lipid dynamics. Freshly laid eggs consist almost exclusively of
reserve, which makes egg production a potentially important
elimination route for lipophyllic compounds. The reserve allocation to
reproduction is via a buffer that comes with species-specific buffer
handling rules. Many aquatic species spawn once a year only (e.g.\
most bivalves and fish), which implies that the buffer size gradually
increases between two spawning events and makes a sharp jump down at
spawning. The body burden can also make a jump at spawning (up or
down, depending on the properties of the compound).

The difference in lipid content between reserve and structure invites
for the application of a nested two-compartment model, where the
exchange with the environment is via structure. This links up nicely
with food uptake, because reserve does not play a role in it, and food
uptake is also proportional to squared structural length. An important
difference with the nested two-compartment model is, however, that the
size of the compartments typically changes in time, especially the
reserve. When redistribution of the compound between the compartments
reserve and structure is relatively fast, and the nested
two-compartment model for the compound reduces to a one-compartment
one, reserve dynamics still affects toxico-kinetics, because the lipid
content is changing in time. The resulting dynamics for active uptake
from food amounts to

\begin{equation}
  \label{eqn:DEB}
  \frac{d} {dt} Q = (P_{Vd} c_d + P_{VX} f c_X) \frac{v_e} {L} 
    - Q(P_{VW} \frac{v_e} {L} + r)
  \quad \mbox{with} \quad
  P_{VW} = 1 + P_{EV} (m_E + m_{ER}) 
\end{equation}
where $c_d$ and $c_X$ are the concentrations of the compound in the
environment and in food, $f$ is the scaled functional response,
$P_{Vd}$ and $P_{VX}$ are the partition coefficients of the compound
in structure and environment or food. The reserve density $m_E$ and
the reproduction buffer density $m_{ER}$ now modify the partition
coefficient between structure and biomass (i.e. reserve plus
structure), via the partition coefficient between reserve and
structure $P_{EV}$. DEB theory specifies how structural length
$L$, the reserve density $m_E$ and the reproduction buffer density
$m_{ER}$ change in time. 

The reproduction buffer is not of importance in all species, and not
always in males. If food density is constant, the reserve density
$m_E$ becomes constant. In those situations the structure-biomass
partition coefficient $P_{VW}$ is constant as well. If also the
dilution by growth can be neglected, i.e.\ $r = 0$ and $L$ is
constant, Eq~\ref{eqn:DEB} still reduces to the one-compartment
model Eq~\ref{eqn:1-comp}.

Many accumulation-elimination experiments are done under starvation
conditions; e.g.\ it is hardly feasible to feed mussels adequately in
the laboratory. The reserve density decreases during the experiment,
so the chemical composition is changing, which can affect the
toxico-kinetics \cite{HareSche94}.

Some situations require more advanced modelling of the uptake and
eliminations route, where e.g.\ gut contents exchanges with the body
in more complex ways, and defecation might be an elimination route.

\subsection{Metabolism and transformation}

Both uptake and elimination can depend on the metabolic activity
\cite{MoleKooy2000}. Respiration is frequently used as a quantifier
for metabolic activity. This explains the popularity of body size
scaling relationships for respiration \cite{Pete83}, and the many
attempts to relate many other quantities to respiration. In the
context of DEB theory, however, and that of indirect calorimetry,
respiration is a rather ambiguous term, because it can stand for the
use of dioxygen, or the production of carbon dioxide or heat. These
are not all proportional to each other, however. Moreover all these
three fluxes have contributions from various processes, such as
assimilation, maintenance, growth etc.  Since the use of reserves
fuels all non-assimilatory activities, this is an obvious quantifier
to link to the rate at which compounds are transformed or taken up.
For uptake of compounds via the respiratory system, the use of
dioxygen might be a better quantity to link to uptake under aerobic
conditions.

Respiration rates turn out to be cubic polynomials in structural
length in DEB theory, which resemble the popular allometric functions
numerically in great detail. The coefficients depend on the
nutritional conditions in particular ways. Since elimination rates
are inversely proportional to length because of surface area-volume
interactions, as has been discussed, and the specific metabolic rate
is very close to this relationship, it is by no means easy to evaluate
the role of metabolism in direct uptake and elimination in undisturbed
subjects.

The role of metabolism is easier to access for elimination via
products and if the elimination rate is not proportional to the
internal concentration, but has a maximum capacity. The classic example
is the elimination of alcohol in human blood \cite{Wagn58}. This type
of kinetics can be described as

\begin{equation}
  \label{eqn:alc}
  \frac{d} {dt} Q = k_e P_{0d} c - k_e Q/(K_Q + Q)
\end{equation}
where $K$ is a half saturation constant for the elimination process.
It reduces to the one-compartment model Eq~\ref{eqn:1-comp} for
small internal concentrations, relative to the half saturation
constant, $K_Q \gg Q$. The elimination rate can now be linked to
metabolic activity, and so to body size. If particular organs are
involved, such as the liver in the case of alcohol, the DEB theory can
be used to study adaptation processes to particular metabolic
functions. In the case of alcohol, the uptake term should obviously be
replaced by a more appropriate one that applies to the particular
subject.

Many toxicants are metabolically modified. This especially applies to
lipophyllic compounds, which are typically transformed into more
hydrophyllic ones, which are more easily excreted but also
metabolically more active. The rate of transformation can be linked to
the metabolic rate, and so depends on body size and nutritional
conditions. These metabolic products can be more toxic than the
original lipophyllic compound.

\section{Bio-availability}

Compounds are not only transformed in the organism, but also in the
environment which affects their availability. Many have an ionic and a
molecular form, which are taken up at different rates; the ionic
species requiring counter ions, which complicates their uptake.
Speciation depends on the concentration of compound and environmental
properties, such as the pH. It can vary in time and also occurs inside
the organism, but the internal pH usually varies within a narrow band
only. Models for mixtures of chemicals can be used in this case (see
section on effects). Internal concentration gradients could develop
if transport inside the organism is slow; film models should be used
in this case. A nice example of a case where concentration gradients
result from transformation in combination with transport is the fluke
\emph{Fasciola} which has an aerobic metabolism near its surface with
the micro-aerobic environment inside its host, but an anaerobic
metabolism in the core of its body \cite{Tiel82}.

Another problem, which occurs especially in soils, is that the
transport through the medium can be slow enough for concentration
gradients to develop around the organism. Film models should then be
used again.

A major problem in the translation of laboratory toxicity tests to field
situations is the formation of ligands with (mainly) organic compound
that are typically abundant in the field, but not in the bioassay.
Ligands reduce the availability substantially, and typically has a
rather complex dynamics. Moreover compounds can be transformed by
(photo)chemical transformation and by actions of (micro)organisms.
This implies that the concentration of available compounds changes in
time, and our methodology to assess effects of chemical compounds
should be able to take this into account.

These processes of transformation require compound-specific modelling
and this short section demonstrates that bio-availability issues
interact with toxico-kinetics and effects of chemical compounds in
dynamic ways, which calls for a dynamic approach to effects of
chemicals \cite{Kooy97,JageHeug2006}.

\section{Effects at the individual level}

Compound affect individuals via changes of parameter values as
functions of the internal concentration
\cite{JageCrom2004,PeryFlam2002}. The parameter values are independent
of the internal concentration in the ``enough''-range of the compound.
This implies the existence of an internal No Effect Concentrations
(NECs) at either end of the ``enough''-range; the upper end is
typically of interest for ecotoxicological applications.  Outside the
``enough''- range the value of the target parameter is approximately a
linear function of the internal concentration as long as the changes
in parameter value are small; the inverse slope is called the
tolerance concentration (a large value means that the compound is not
very toxic). Small changes in parameter values do not necessarily
translate into small changes of some end-point, such as the cumulative
number of offspring \cite{KooyBeda96b} or the body size
\cite{KooyBeda96c} at the end of some standardised exposure period.
DEB theory specifies how exactly changes in parameter values translate
into the performance of the individual. Typical target parameters are
the specific maintenance costs, or the yield of structure on reserve,
or the maximum specific assimilation rate or the yield of reserve on
food or the yield of offspring on reserve. For effects on survival,
the hazard rate serves the role of target parameter, and the inverse
``tolerance concentration'' for the hazard rate is called the killing
rate. Mutagenic compounds can induce tumours \cite{LeeuZonn2001}, but
also accelerate ageing by enhancing the effects of ROS.  The
partitioning fraction for mobilised reserve can be the target
parameter for endocrine disruptors.

Using sound theory for how effects depend on internal concentrations,
DEB-based theory can handle varying concentrations of toxicant
[37-40]\nocite{PeryBeda2001,AldaJage2006,KlepBeda97,KlepBeda97a}, even
pulse exposures \cite{PietJage2006}. DEB theory applies to all
organisms, including bacteria that decompose organic pollutants. A
proper description of this process should account for adaptation
\cite{BranKelp2004}, co-metabolism \cite{BranLeeu2003} and the fact
most bacteria occur in flocculated form in nature \cite{BranKooy2000},
which affect the availability of the compound.

The model of linear effects of internal concentrations on parameter
values has been extended into several directions, such as adaptation
to the compound, inclusion of the recent exposure history via receptor
dynamics \cite{JageKooy2005} and attempts to include particular
molecular mechanisms \cite{MullNisb97}.

\subsection{Mixtures and NECs}

Mixtures of compounds affect parameters values via addition of the
effects of single compounds, plus an interaction term which is
proportional to the products of the internal concentrations of the
compound \cite{BaasHout2007}. This interaction term can be positive or
negative; a construct that is the core of the analysis of variance
(ANOVA) model and rests on a simple Taylor series approximation of a
general non-linear multivariate function, which only applies for small
changes of parameter values; the non-linearities of the effects should
be taken into account for larger changes. These non-linearities might
well be specific for the compound and the species and, therefore, lack
generality. Notice that linear effects on parameter values translate
into non-linear effects of the performance of the individual because
the DEB models are non-linear. Also notice that each DEB parameter has
a NEC value for any compound; the lowest value among all parameters
might be considered as \emph{the} NEC of a compound for the organism,
but its estimation requires to study effects on all 13 parameters of
the standard DEB model, in principle. Since this study can be demanding,
it is in practice essential to talk about the NEC of a compound for
an organism \emph{for a particular DEB parameter}.

The NEC reflects the ability of the individual to avoid changes of
performance. From a statistical point of view, this robust parameter
has very nice properties
[48-50]\nocite{AndeBeda2000,BaasJage2008,KooyBeda96a}.  The NEC is
\emph{not} meant to imply that some molecules of a compound don't have
an effect, while other molecules do. The removal of a kidney in a
healthy person can illustrate the NEC concept: the removal implies an
effect at the sub-organismic level, but this effect generally does not
translate into an effect at the individual level.  The NEC, therefore,
depends on the level of observation. We can delineate three cases of
how compounds in the mixture combine for the NEC
\begin{itemize}
\item the presence of other compounds is of no relevance to the NEC of
  any particular compound
\item the various compounds add, like they do for effects, and at the
  moment effects show up, the amounts of the compounds that show no
  effects remain constant
\item the various compounds add, like they do for effects, and the
  amounts of the compounds that show no effects continue changing with
  the internal concentration of the compounds; if a compound continues
  accumulation more than other compounds, its NEC increases while
  that of the other compounds in the mixture decreases
\end{itemize}
The third case is possibly most realistic, but also computationally
the most complex. In many practical situations the results are very
similar to the second case, which can be used as an approximation. If
compounds in a mixture are equally toxic, and so all have the same
NEC, the second case is formally identical to this special case
\cite{JageBaas2008}. This way of described effects of mixtures turns
out to fit well with experimental data \cite{BaasHout2007} and each
pair of compounds have a single interaction parameter, which does not
change in time. If there are $k$ compounds in a mixture, there are $k
(k - 1)/ 2$ interaction parameters, just like in ANOVA.

A further reflection on the NEC might clarify the concept. Any
compound affects (in principle) all DEB parameters (including the
hazard rate), but the NEC for the various parameters differ. This
makes that, if the internal concentration increases, the parameter
with the lowest NEC first starts to change, but other parameters
follow later. In a mixture of compounds, this can readely lead to a
rather complex situation where in a narrow range of (internal)
concentrations of compounds in a mixture several parameters start
changing \cite{JageBaas2008}. Even in absence of the above-mentioned
chemical interactions of compounds on a single parameter, interactions
via the energy budget occur, which are hard to distinguish from the
chemical interactions on a single parameter. Chemical interactions are
typically rare, but interactions via the budget always occur. 

\subsection{Hormesis}

Hormesis, the phenomenon that low concentrations of a toxicant seem to
have a stimulating rather than an inhibiting effect on some endpoint,
can result from interactions of the compound with a secondary stress,
such as resulting from very high levels of food availability. If a
compound decreases the yield of structure on reserve, it reduces
growth and delays birth (if an embryo is exposed) and puberty (in the
case of juveniles), but also reduces the size at birth. A reduction of
growth indirectly reduces reproduction, because food uptake is linked
to size. Since it also reduces size at birth, the overall effect can
be a hormesis effect on reproduction (in terms of number of offspring
per time) \cite{Kooy2008}. Indirect effects on reproduction differ
from direct effects by not only reducing, but also delaying
reproduction. This has important population dynamical consequences.

\subsection{Co-variation of parameter values}

A very powerful property of the standard DEB model and the
one-compartment model is that they imply rules for how parameter
values co-vary among species and compounds
\cite{Kooy86a,KooyBaas2007,SousDomi2008}; this variation directly
translates into how expected effects vary. These expectations can be
used to fill gaps in knowledge about parameter values, but cannot
replace the need for this knowledge. Evolutionary adaptations and
differences in mode of action of compounds can cause deviation from
expected parameter values for species, and compounds, respectively.

The reasoning behind the scaling relationships for the standard model
rests on the assumption that parameters that relate to the local
biochemical environment in an organism are independent of the maximum
body size of a species, but parameters that relate to the physical
design of an organism depends on the maximum size. Strange enough,
this simple assumption fully species the covariation of parameter
values. The application is best illustrated with the maximum length
$L_m$ an endotherm can reach in the standard DEB model. This length is
a simple function of three parameters, $L_m = \kappa \{p_{Am}\}/
[p_M]$, where $\kappa$ is the fraction of mobilised reserve that is
allocated to the soma, $\{p_{Am}\}$ is the surface-area specific
maximum assimilation energy flux and $[p_M]$ is the volume-specific
somatic maintenance cost. Since $\kappa$ and $[p_M]$ depend on the
local biochemical environment, they are independent of maximum length,
which implies that $\{p_{Am}\}$ must be proportional to maximum
length. All other parameters can be converted in simple ways to
quantities that depend on the local biochemical environment; these
transformations then defined how they depend on maximum length. When
we divide the maturity at birth and puberty by the cubed maximum
length, we arrive at a maturity-density, which reflects the local
biochemical environment and should not depend on maximum length. So
the maturity at birth and puberty are proportional to the cubed
maximum length. Many quantities, such as the use of dioxygen by an
individual, can be written as functions of parameter values and
amounts of reserve and structure. So the maximum respiration rate of a
species is a function of parameter values, while we know of each
parameter how it depends on maximum length. It can be show that
maximum respiration rate scales between a squared and a cubed maximum
length, and the weight-specific respiration with weight to the power
$-1/4$, a well-known result since Kleiber \cite{Klei32}.

The reasoning behind the scaling relationships for the one-compartment
model rests on the assumption that transport to and from the
compartment is skewly symmetric \cite{KooyJage2004}. The ratio of the
concentrations in the compartment and the environment at equilibrium
is a ratio of uptake and the elimination rates, just like the maximum
length of the individual in the standard DEB model. This implies, see
\cite{KooyJage2004}, that the uptake rate is proportional to the
square root of the partition coefficient, and the elimination rate is
inversely proportional to the square root of the partition
coefficient. Film models are extensions of the one-compartment model,
that behave at the interface between the environments basically in the
same way as an one-compartment model; only around this interface they
differ because film models account for concentration gradients. This
deviation can be taken into account with the result that elimination
rates are (almost) independent of the partition coefficient for low
values of the partition coefficient and inversely proportional to it
at high values. Effects parameters can be included into the scaling
reasoning is similar ways, with the result that the NEC is inversely
proportional to the partition coefficient, and the tolerance
concentration or the killing rate is proportional to the partition
coefficient.

The octanol-water partition coefficient is frequently taken as a
substitute for the body-water partition coefficient, with the
advantage that reliable computational methods exist to evaluate this
partition coefficient from the chemical structure of the molecule. In
practice, however, octanol is not an ideal chemical model for
organisms which change the chemical composition of their bodies. This
makes that the co-variation of NECs, elimination and killing rates
show less scatter and can be expected on the basis of the variation
between each of these three parameters and the octanol-water partition
coefficient \cite{KooyBaas2007}.

We are unaware of any descriptive model for toxico-kinetics and/or
metabolic organisation for which theory on the co-variation of
parameter values is available, and we doubt that it even possible to
derive such theory for descriptive models. Co-variation theory is not
available for the so-called net-production models \cite{LikaKooy2003},
for instance, where maintenance needs are first subtracted from
assimilation before allocation to storage, growth or reproduction. The
fact that the predicted relationships of now over thirty
eco-physiological quantities, such as length of the embryonic and
juvenile periods, maximum reproduction rate, maximum growth rate,
maximum population growth rate, vary with the maximum body size of
species in ways that match empirical patterns provides strong
support of the DEB theory. The one-compartment and standard DEB models
share the property that the independent variable (the partition
coefficient in the case of toxicokinetics and the maximum length in
the case of budgets) can be written as a ratio of an incoming flux (of
toxicant and food, respectively) and an outgoing flux (excretion and
maintenance, respectively). This shared property seems to be crucial
for the core theory.

One of the many possible applications of the scaling relationships is
in the effects of mixtures of compounds with similar modes of action,
such as the poly-chlorinated hydrocarbons. Suppose we know the
concentrations and the partition coefficients of the compounds in the
mixture. We then link the elimination rates, the NECs and the
tolerance concentrations to the partition coefficients in the way
described, and estimate the three proportionality constants for the
results of a bioassays with the mixture.

The sound theoretical basis for effects of toxicants in combination
with rules for the co-variation of parameter values offers the
possibility for extrapolation, from one individual to another, from
one species of organism to another, and, sometimes, from one type of
compound to another \cite{KooyBaas2007}. These crosslinks partly
reduce the need for a huge experimental effort that should be invested
in more advanced forms of environmental risk assessment, such as
discussed in Brack \emph{et al.} \cite{BracBakk2005}. Moreover, the
theory simplifies to parameter poor models under particular
conditions. It has been demonstrated that many popular empirical
models turn out to be special cases of the general theory
\cite{Kooy2000}. This might help in particular applications.

\section{Effects at the population and ecosystem level}

At high food levels, organisms grow and reproduce fast and the
maintenance costs comprise only a tiny fraction of the budget of the
individuals. If a compound increases the maintenance cost for
individuals, say by a factor two, these effects are hardly felt by a
fast-growing population. Fast growth never lasts long in nature, due
to the depletion of food resources. At carrying capacity, where the
generation of food resources just matches the maintenance needs of a
population (this is the maintenance needs of the collection of
individuals plus a low reproductive output that cancels the
mortality), maintenance costs comprise the dominant factor of the
budget of individuals. If a toxic compounds now increases the
maintenance cost by a factor two, it in fact reduces the carrying
capacity by a factor two. This simple argument shows that the effects
of toxicants on populations is dynamic, even if the concentration of
the compound would be constant \cite{Kooy85a}. It also shows that no
single quantifier for toxicity can exist at the population level.

If a toxic compound increases the cost of growth or reproduction, the
effects hardly depend on the growth rate of the population, so
on the food level, which shows that the mode of action is important
for how effects on individuals translate to those on populations. It
might be difficult to tell the various modes of action apart on the
basis of the results from a standardised toxicity bioassay. The reason
why the mode of action is still important is in the biological
significance of the observed effect, which must be found at the
population level. Details in the reproduction strategy of populations
turn out to be important for how effects on reproduction translate to
the population level \cite{AldaJage2005}.

Although bioassays with meso-cosms have the charm of being close to the
actual interest of effects of toxicants to be avoided, the
experimental control is extremely weak which results in a huge scatter
of trajectories of experimental meso-cosms. The result is that the
effects have to be huge to recognise them as effects \cite{Kooy88a}.
Moreover the expected long term behaviour of chemically perturbated
ecosystems is very complex, as shown by bifurcation analysis
\cite{KooiBont2008}. 

The specific population growth rate integrates the various
performances of individuals naturally, and can rather easily be
evaluated \cite{KooyHans96}. A delay of the onset of reproduction can
be at least as important as a reduction of the reproduction for the
fate of the population.

\section{Concluding remarks}

We argued that models for effects of chemical compounds should have
three modules:
\begin{itemize}
\item dynamic energy budgets for how organisms generally deal with
  resource uptake and allocation
\item toxico-kinetics for how organisms acquire the compounds
\item chances of budget parameters as function of the internal concentrations
\end{itemize}
We discussed the basics for each of these modules: the standard DEB
model, the one-compartment model and the linear change in target
parameters. We also indicated were and how these models can be
extended, from simple to more complex, to include particular
phenomena. We discussed how budgets affect both the kinetics and the
effects and, therefore, have a central role in effects models.
Practice teaches that the restriction of realistic modelling is not in
the model formulation as such, but in the useful application of these
models to data. More complex models have more parameters and many of
these parameters are by no means easy to extract from available
experimental data. They require knowledge of physiological and
ecological processes that are typically outside the scope of typical
(eco)toxicological research. Kooijman \emph{et al.}
\cite{KooySous2008} discusses why any particular application of DEB
theory requires only a limited set of parameter values, and how these
values can be obtained from simple observation on growth and
reproduction at several levels of food availability.

The practical need to fill in gaps in knowledge about parameter values
is the reason why due attention has been given to theory for the
co-variation of parameter values; this theory naturally follows from
the logical structure of the one-compartment model and the standard
DEB model. Extensions of both models can modify the co-variation of
parameters, as has been discussed.

Contrary to descriptive models, models with strong links with
underlying processes can be used for a variety of extrapolation
purposes, from acute to chronic exposure, from one species to another,
from one compound to another, from individuals to populations, from
laboratory to field situations \cite{JageHeug2006}. Such
extrapolations are typically required in environmental risk assesment,
where NECs should play a key role \cite{KooyBeda96d}. The use of
models to predict exposure in the environment is frequent, but to
predict effects is still rare. The complexity of the response of
organisms to changes in their chemical environment doubtlessly
contributed to this. Yet we think that thirty years of applications of
DEB theory to quantify effects of compounds on organisms have
demonstrated that the theory is both effective and realistic. Many of
the computations behind the models in this chapter can be done with
the freely downloadable software package DEBtool:
http://www.bio.vu.nl/thb/deb/deblab/

\section{Acknowledgements}

This research has been supported financially by the European Union
(European Commission, FP6 Contract No. 003956 and No 511237-GOCE).

{\footnotesize 
%\bibliographystyle{unsrt}
%\bibliography{kinetics}
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\end{thebibliography}

}
\end{document}

Reviewers:

Dr. Cedric Bacher      Cedric.Bacher@ifremer.fr
Institut Francais de Recherche pour l'exploitation de la mer - IFREMER
CREMA CNRS-IFREMER
BP 5 PLACE DU SEMINAIRE
17137 l'HOUMEAU
FRANCE

Prof. Jan Hendriks     a.j.hendriks@science.ru.nl
Faculteit der Natuurwetenschappen Wiskunde en Informatica
Institute for Wetland and Water Research
Radboud Universiteit
Comeniuslaan 4 6525 HP Nijmegen 

Dr. Hans Lokke         hlo@dmu.dk
Director of Terrestrial Department,
NERI, Vejlsovej 25, DK-8600 Silkeborg, Denmark

Dr. Jaap vd Meer     meer@nioz.nl
Royal Netherlands Institute for Sea Research (NIOZ)
P.O. Box 59, NL-1790 AB Den Burg (Texel) The Netherlands

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