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Dose-response relationships
Tjalling Jager
Theoretical Biology
Dose-response analysis
This morning:
1. Introduction in effects assessment
2. Analysis of survival data
3. Analysis of continuous data
4. Problems with these methods
5. An alternative approach
Why effects assessment?
How toxic is chemical X?– for RA of the production or use of X– for ranking chemicals (compare X to Y)– for environmental quality standards
Need measure of toxicity that is:– good indicator for environment– comparable between chemicals
Test organisms (aquatic)
Standardisation
Toxicity tests are highly standardised (OECD, ISO, etc.):– species– exposure time– endpoints– test medium, temperature etc.
Types of tests
‘Acute’ – short-term– usually mortality or immobility– quantal or discrete response
‘Chronic’– long-term– usually sub-lethal endpoint– graded or continuous response
Standard test set-up
Survival test
Survival test
After 2 days …
Reproduction test
Reproduction test
After 21 days …
Range of Concentrations
Plot response vs. doseR
esp
on
se
log concentration
What pattern to expect?What pattern to expect?
Linear?R
esp
on
se
log concentration
Threshold, linear?R
esp
on
se
log concentration
Threshold, curve?R
esp
on
se
log concentration
S-shape?R
esp
on
se
log concentration
Hormesis?R
esp
on
se
log concentration
Essential chemical?R
esp
on
se
log concentration
Contr.
Standard approaches
NOEC
Res
po
nse
log concentration
LOEC
*
assumes threshold
1. Statistical testing2. Curve fitting
Standard approaches
EC50
Res
po
nse
log concentration
usually no threshold
1. Statistical testing2. Curve fitting
Standard summary statistics
NOEC highest tested concentration where effect is
not significantly different from control
EC50 or LC50 the estimated concentration for 50% effect,
compared to control
Dose-response analysis
This morning:
1. Introduction in effects assessment
2. Analysis of survival data
3. Analysis of continuous data
4. Problems with these methods
5. An alternative approach
Available data
Number of live animals after fixed exposure period Example: Daphnia exposed to nonylphenol
mg/L 0 h 24 h 48 h
0.004 20 20 20
0.032 20 20 20
0.056 20 20 20
0.100 20 20 20
0.180 20 20 16
0.320 20 13 2
0.560 20 2 0
Plot dose-response curve
Procedure– plot fraction survival after 48 h– concentration on log scale
Objective– derive LC50– (seldom NOEC)
0
20
40
60
80
100
0.001 0.01 0.1 1
concentration (mg/L)
su
rviv
al
(%)
first: parametric analysisfirst: parametric analysis
What model?
Requirements– start at 100% and decrease to zero– inverse cumulative distribution?
0
20
40
60
80
100
0.001 0.01 0.1 1
concentration (mg/L)
su
rviv
al
(%)
Cumulative distributions
E.g. the normal distribution …
prob
abili
ty d
ens
ity
cum
ulat
ive
den
sity
1
Distribution of what?
Assumptions– animal dies instantly when exposure exceeds ‘threshold’– threshold varies between individuals– spread of distribution indicates individual variation
pro
bab
ility
de
nsity
cum
ulat
ive
de
nsity
1
pro
bab
ility
de
nsity
pro
bab
ility
de
nsity
cum
ulat
ive
de
nsity
1
cum
ulat
ive
de
nsity
cum
ulat
ive
de
nsity
1
Concept of “tolerance”
0
20
40
60
80
100
0.001 0.01 0.1 1
concentration (mg/L)
su
rviv
al
(%)
0
20
40
60
80
100
0.001 0.01 0.1 1
concentration (mg/L)
su
rviv
al
(%)
1
cum
ulat
ive
dens
itycu
mul
ativ
e de
nsity
1
pro
bab
ility
de
nsity
pro
bab
ility
de
nsity
20% mortality
20% mortality
What is the LC50?
0
20
40
60
80
100
0.001 0.01 0.1 1
concentration (mg/L)
su
rviv
al
(%)
0
20
40
60
80
100
0.001 0.01 0.1 1
concentration (mg/L)
su
rviv
al
(%)
1
cum
ulat
ive
dens
itycu
mul
ativ
e de
nsity
1
pro
bab
ility
de
nsity
pro
bab
ility
de
nsity
50% mortality
50% mortality
?
Graphical method
Probit transformation
2 3 4 5 6 7 8 9probits
std. normal distribution + 5
Linear regression on probits versus log concentration
0
20
40
60
80
100
0.001 0.01 0.1 1
concentration (mg/L)
0
20
40
60
80
100
0.001 0.01 0.1 1
data
mo
rta
lity
(%
)
Fit model, least squares?
0
20
40
60
80
100
0.001 0.01 0.1 1
concentration (mg/L)
surv
ival
(%
)
Error is not normal:– discrete numbers of survivors– response must be between 0-100%
Error is not normal:– discrete numbers of survivors– response must be between 0-100%
How to fit the model
Result at each concentration as binomial trial Probability to survive is p, to die 1-p Predicted p = f(c) Estimate parameters of the model f
– maximum likelihood estimation– weighted least-squares … – chi-square for goodness of fit …
11
Fit model, least squares?
0
20
40
60
80
100
0.001 0.01 0.1 1
concentration (mg/L)
surv
ival
(%
)
Max. likelihood estimation
0
20
40
60
80
100
0.001 0.01 0.1 1
concentration (mg/L)
surv
ival
(%
)
Which distribution?
Popular distributions– log-normal (probit)– log-logistic (logit)– Weibull
ISO/OECD guidance document
A statistical regression model itself does not have any meaning, and the choice of the
model is largely arbitrary.
A statistical regression model itself does not have any meaning, and the choice of the
model is largely arbitrary.
Resulting fits: close-up
10-1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
concentration
fra
ctio
n s
urv
ivin
g
datalog-logisticlog-normalWeibullgamma
LC50 -log lik.
Log-logistic 0.225 16.681
Log-normal 0.226 16.541
Weibull 0.242 16.876
Gamma 0.230 16.582
Non-parametric analysis
Spearman-Kärber: wted. average of midpoints
0
20
40
60
80
100
0.001 0.01 0.1 1
log concentration (mg/L)
surv
ival
(%
)
weights is number of deaths in interval
only for symmetrical distributions
weights is number of deaths in interval
only for symmetrical distributions
“Trimmed” Spearman-Kärber
0
20
40
60
80
100
0.001 0.01 0.1 1
log concentration (mg/L)
surv
ival
(%
)
Interpolate at 95%
Interpolate at 5%
Summary: survival
Survival data are quantal data, reported as fraction responding individuals
Analysis types– parametric (tolerance distribution)– non-parametric (trimmed Spearman-Kärber)
Model hardly affects LC50
Error is ‘multinomial’
Dose-response analysis
This morning:
1. Introduction in effects assessment
2. Analysis of survival data
3. Analysis of continuous data
4. Problems with these methods
5. An alternative approach
Difference graded-quantal
Quantal: fraction of animals responding– e.g. 8 out of 20 = 0.4– always between 0% and 100%– no standard deviations
Graded: degree of response of the animal– e.g. 85 eggs or body weight of 23 g– usually between 0 and infinite– standard deviations when >1 animal
Analysis of continuous data
Endpoints– In ecotoxicology, usually growth (fish) and
reproduction (Daphnia)
Two approaches– NOEC and LOEC (statistical testing)– ECx (regression modelling)
Derive NOEC
NOEC
Res
po
nse
log concentration
Contr.
LOEC
*
Derivation NOEC
ANOVA: are responses in all groups equal? H0: R(1) = R(2) = R(3) …
Post test: multiple comparisons to control, e.g.:– t-test with e.g. Bonferroni correction– Dunnett’s test– Fisher’s exact test with correction– Mann-Whitney test with correction
Trend tests – stepwise: remove highest dose until no sign. trend is left
What’s wrong?
Inefficient use of data (most data are ignored) No statistically significant effect does not
mean no effect– large effects (>50%) may occur at the NOEC– large variability leads to high NOECs
However, NOEC is still used!
NOECNOEC
Re
sp
on
se
log concentration
Contr.Contr.
LOEC
*LOECLOEC
*
See e.g., Laskowski (1995), Crane & Newman (2000)
Regression modelling
Select model– log-logistic (ecotoxicology)– anything that fits (mainly toxicology)
• straight line• exponential curve• polynomial
Re
sp
on
se
log concentration
Re
sp
on
se
log concentration
Least-squares estimation
concentration (mg/L)
0
20
40
60
80
100
0.001 0.01 0.1 1
rep
rod
uct
ion
(#e
gg
s)
n
iii estRmeasRSSQ
1
2.)(.)(
n
iii estRmeasRSSQ
1
2.)(.)(
Note: lsq is equivalent to max. likelihood, assuming normally-distributed errors
Note: lsq is equivalent to max. likelihood, assuming normally-distributed errors
Example: Daphnia repro test
Standard protocol– take juveniles <24 h old– expose to chemical for 21 days– count number of offspring daily– use total number of offspring after 21 days– calculate NOEC and EC50
Example: Daphnia and Cd
NOEC is (probably) zero
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
10
20
30
40
50
60
70
80
90
100
concentration
# ju
v./f
emal
e
Example: Daphnia repro
Put data on log-scale and fit sigmoid curve
10-2
10-1
100
101
0
10
20
30
40
50
60
70
80
90
100
concentration
# ju
v./f
emal
eEC10
0.13 mM(0.077-0.19)
EC50 0.41 mM
(0.33-0.49)
Regression modelling
Advantage– use more of the data– ECx is estimated with confidence interval– poor data lead to large confidence intervals
Model is purely empirical– no understanding of the process– extrapolation is dangerous!
10-2
10-1
100
101
0
10
20
30
40
50
60
70
80
90
100
concentration
# ju
v./f
em
ale
10-2
10-1
100
101
0
10
20
30
40
50
60
70
80
90
100
concentration
# ju
v./f
em
ale
EC100.13 mM
(0.077-0.19)
EC100.13 mM
(0.077-0.19)
EC500.41 mM
(0.33-0.49)
EC500.41 mM
(0.33-0.49)
Summary: continuous data
Repro/growth data are ‘graded’ responses– look at average response of animals– not fraction of animals responding!
Thus: no ‘tolerance distribution’!
Analysis types– statistical testing (e.g., ANOVA) NOEC– regression (e.g., log-logistic) ECx
Dose-response analysis
This morning:
1. Introduction in effects assessment
2. Analysis of survival data
3. Analysis of continuous data
4. Problems with these methods
5. An alternative approach
Problems
Dilemma of risk assessment
Protection goalAvailable data
• different exposure time • different temperature• different species• time-variable conditions• limiting food supplies• interactions between species• …
Extrapolation?
single time pointsingle endpoint
Available data Assessment factor
Three LC50s 1000
One NOEC 100
Two NOECs 50
Three NOECs 10
‘Safe’ level for field system
LC50ECx
NOECRes
po
nse
log concentration
Where’s the science?
No attempt to understand process of toxicity Dose-response approaches are descriptive Extrapolation through arbitrary ‘assessment factors’ Ignores that LC50/ECx/NOEC change in time
10Three NOECs
50Two NOECs
100One NOEC
1000Three LC50s
Assessment factor
Available data
10Three NOECs
50Two NOECs
100One NOEC
1000Three LC50s
Assessment factor
Available data
LC50ECx
NOECRes
po
nse
log concentration
LC50ECx
NOECRes
po
nse
log concentration
Res
po
nse
log concentration
Effects change in time
0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
concentration
fra
cti
on
su
rviv
ing
24 hours
48 hours
LC50 s.d. tolerance
24 hours 0.370 0.306
48 hours 0.226 0.267
Toxicokinetics
Why does LC50 decrease in time? Partly:– effects are related to internal concentrations– accumulation takes time
time
inte
rna
l c
on
ce
ntr
ati
on
time
inte
rna
l c
on
ce
ntr
ati
on
chemical A
chemical B
chemical C
small fish
large fish
Daphnia
Change in timedepends on1. chemical2. test species
Change in timedepends on1. chemical2. test species
Chronic tests
With time, control response increases and all parameters may change …
10-2
10-1
100
101
0
10
20
30
40
50
60
70
80
90
100
concentration
# ju
v./f
emal
eincreasing time (t = 9-21d)
EC10 in time
0.5
1
1.5
2
2.5
0 5 10 15 200
survival
body length
cumul. reproductioncarbendazim
Alda Álvarez et al. (2006)
time (days)0 2 4 6 8 10 12 14 16
0
20
40
60
80
100
120
140
pentachlorobenzene
time (days)
Toxicity is a process in time
Effects change in time, how depends on:– endpoint chosen– species tested– chemical tested
Ignored by standardising exposure time
No such thing as the ECx/LC50/NOEC– difficult to compare chemicals, species, endpoints
Dose-response analysis
This morning:
1. Introduction in effects assessment
2. Analysis of survival data
3. Analysis of continuous data
4. Problems with these methods
5. An alternative approach
Biology-based modelling
Make explicit (but simple) assumptions on mechanisms of toxicity
toxico-kinetics
toxico-dynamics
internalconcentration
in timeexternal
concentration(in time)
effectsin time
Toxicokinetics
Simplest form: 1-compartment model More detail in Module 2 …
time
inte
rnal
co
nce
ntr
atio
n
elim
inatio
n ra
te
?
Why do animals die?
Instant death at certain threshold?
Newman & McCloskey (2000)
lethalexposure
lethalexposure
?
Hazard modelling
Chemical increases probability to die
internal concentration
haza
rd r
ate
internal concentration
hazard rate
survival in time
Effect depends on internal concentration
1 comp.kinetics
blank value
NEC
Example DEBtox
Results
Parameters are • time-independent• comparable between species and chemicals
Use parameters to predict effects• on different time-scale• of time-varying exposure• of different size animals• of different chemicals• …
Sub-lethal effects
Sub-lethal effects
toxicant
Sub-lethal effects
Sub-lethal effects
Dynamic Energy Budgets
growth
reproduction
assimilation
maintenance
growth and repro in time
DEBtox basics
internal concentration
DE
B p
aram
eter
NEC
blank value
internal concentration
DE
B p
aram
eter
NEC
blank value
DEB
toxicokinetics
Effect depends on internal concentration Chemical changes parameter in DEB model
Example DEBtox
Results
Parameters are • time-independent• comparable between species and chemicals
Use parameters to predict effects• on different time-scale• of time-varying exposure• of different size animals• at population level• …
Life-cycle data
Follow growth/repro/survival over large part of the life cycle
Alda Álvarez et al. (2006)
Example:– nematode Acrobeloides nanus– exposed to cadmium in agar for
35 days– body size, eggs and survival
determined regularly
Example: A. nanus and Cd
0 5 10 15 20 25 30 35
20
30
40
50
60
time
bo
dy
len
gth
0 5 10 15 20 25 30 35
20
30
40
50
60
time
bo
dy
len
gth
0 5 10 15 20 25 30 350
100
200
300
timec
um
ula
tiv
e o
ffs
pri
ng
0 5 10 15 20 25 30 350
100
200
300
timec
um
ula
tiv
e o
ffs
pri
ng
0 5 10 15 20 25 30 350
0.2
0.4
0.6
0.8
1
time
fra
cti
on
su
rviv
ing
0 5 10 15 20 25 30 350
0.2
0.4
0.6
0.8
1
time
fra
cti
on
su
rviv
ing
Alda Álvarez et al. (2006)
Mode of action: costs for growthParameters:
7 for basic life history7 for chemical behaviour
Mode of action: costs for growthParameters:
7 for basic life history7 for chemical behaviour
Alternative approach
Biology-based methods (DEBtox)– make explicit assumptions on processes– analyse all data in time– parameters do not change in time– basis for extrapolations
toxico-kinetics
toxico-dynamics
internalconcentration
in timeexternal
concentration(in time)
effectsin time
externalconcentration
(in time)
effectsin time
Summary
Remember
Survival Usually acute
Growth / repro Usually (sub)chronic
Remember
Survival Usually acute Quantal response (dead
or alive)
Growth / repro Usually (sub)chronic Graded response
(#eggs, size)
Remember
Survival Usually acute Quantal response (dead
or alive) Needs at least 10
animals per dose
Growth / repro Usually (sub)chronic Graded response
(#eggs, size) Needs 1 animal per
dose (more for NOEC)
Remember
Survival Usually acute Quantal response (dead
or alive) Needs at least 10
animals per dose Analyse by finding
tolerance distribution or non-parametric
Growth / repro Usually (sub)chronic Graded response
(#eggs, size) Needs 1 animal per
dose (more for NOEC) Analyse by standard
regression techniques (curve fitting)
Remember
Survival Usually acute Quantal response (dead
or alive) Needs at least 10
animals per dose Analyse by finding
tolerance distribution or non-parametric
LC50, EC50 …
Growth / repro Usually (sub)chronic Graded response
(#eggs, size) Needs 1 animal per
dose (more for NOEC) Analyse by standard
regression techniques (curve fitting)
NOEC, EC50, EC10 …
Watch out!
Problems with standard analyses– descriptive, no understanding of process– statistics depend on exposure time
Alternative: biology-based– make assumptions on mechanisms– analyse effects data in time
Standard analysis may have role in risk assessment but …
Science needs BB methods
0
20
40
60
80
100
0 0.05 0.1 0.15 0.2 0.25
Cd concentration (mg/L)
tota
l ju
ven
iles
afte
r 15
d
high food
low food
EC50
Data Heugens et al. (2006)
Does food limitation increase effect of cadmium?
Food limitation
growth
reproduction
assimilation
maintenance
ad libitum
5%
Food limitation
growth
reproduction
assimilation
limiting
maintenance
50%
Electronic DEB laboratory
DEBtox– Windows version 2.0.2. (2007)– data from standard tests
Free downloads fromhttp://www.bio.vu.nl/thb/deb/deblab/
DEBtool– open source (Octave, MatLab)– full range of DEB research – advanced DEBtox applications
