Transcript ecotox_course_2016x

Dose-response analysis
Tjalling Jager
Contents
Introduction
 Why and how of ‘dose responses’
 How to fit a curve to data points (statistics)
Current practice
 Analysis of survival data
 Analysis of continuous data
Critical notes
 Limitations of current practice
 Outline of alternatives
Why toxicity testing?
How toxic is chemical X?
– RA of production or use of X
– ranking chemicals (compare X to Y)
– environmental quality standard of X
Need measure of toxicity that is:
– good indicator for (no) effects in the field
– comparable between chemicals
Scientific interest:
– how do chemicals affect organisms?
– stress organism to reveal how they work …
Toxicity is a process
Dynamic interaction organism↔chemical
Response to toxicant depends on
–
–
–
–
–
toxicant
organism
endpoint (trait)
exposure duration/pattern
…
Practical solution: standardisation
– standard test protocols of OECD, ISO, ASTM etc.
– required for risk assessment
– scientific studies also use these protocols
Standard test species (aquatic)
Green algae, diatoms,
cyanobacteria
population growth
Daphnia sp. (magna)
immobility
reproduction
‘Well-documented’ species
mortality
growth
life cycle
Daphnia reproduction test
50-100 ml of welldefined test
medium, 18-22°C
Daphnia reproduction test
Daphnia magna
Straus, <24 h old
Daphnia reproduction test
Daphnia magna
Straus, <24 h old
Daphnia reproduction test
wait for 21 days, and
count total offspring …
Daphnia reproduction test
at least 5 test concentrations in
geometric series …
Response
Response vs. dose
log concentration
Response vs. dose
1. Statistical testing
Contr.
Response
NOEC
*
LOEC
log concentration
What’s wrong?
Inefficient use of data
– most data points are ignored
– NOEC has to be one of the test concentrations
Awkward use of statistics
– absence of proof is no proof of absence …
– level of effect at NOEC regularly >20%
– poor testing leads to high (unprotective) NOECs
But, NOEC is still used …
Contr.
Response
NOEC
*
LOEC
log concentration
Response vs. dose
Response
1. Statistical testing
2. Curve fitting
EC50
What curve to use?
log concentration
Response
Linear?
log concentration
Response
Threshold, linear?
log concentration
Response
Threshold, curve?
log concentration
Response
S-shape?
log concentration
Response
Hormesis?
log concentration
Which curve?
Popular: S-shaped curves
Usually inverse cumulative probability distributions
– continuous, monotonic …
– e.g., log-normal, log-logistic, Weibull
cumulative density
probability density
1
log concentration
log concentration
Which curve?
Popular: S-shaped curves
Usually inverse cumulative probability distributions
– continuous, monotonic …
– e.g., log-normal, log-logistic, Weibull
1- cumulative density
probability density
1
log concentration
log concentration
Summary of why and how
 Aim: quantify/summarise ‘the toxicity’ of a chemical
 Toxicity tests are highly standardised
 Classic ways to extract summary statistic:
– hypothesis testing (e.g., NOEC)
– curve fitting (e.g., EC50)
 S-shaped curves dominate
– choice is quite arbitrary
Introduction 2
How to fit a curve to data
Problem
parameters
dose-resp.
model
predicted
effect
optim.
method
‘error’
model
 How to find ‘best’ parameter values?
 How certain are we of those?
data
Example: growth
 Springtail Folsomia
candida
– cohort followed over time
– weight at t=0 destructive
Time (days)
Weight (µg)
0
1.8*
16
73.0
23
165
30
226
37
242
44
250
51
263
Make a plot
350
300
Body weight (µg)
250
200
150
100
50
0
-50
0
10
20
30
Time (days)
40
50
60
Add a model
 Von Bertalanffy model for body weight
– find ‘best’ values for parameters Wm and rb
350
300
Body weight (µg)
250
200
150
100
50
0
-50
0
10
20
30
Time (days)
40
50
60
Residuals
 Can model go through all points exactly?
– generally not
 Why not?
– measurement error
– biological variation
– the model is always ‘wrong’
 That’s true, but let’s assume ...
–
–
–
–
model is completely right
data points are independent samples
error is normally distributed
same variation for all points (homoscedasticity)
error model →
least squares
Model fitting
 Minimise the squared residuals
– model for weight, fix W0
parameters
350
dose-resp.
model
300
Body weight (µg)
data
‘error’
model
250
predicted
effect
200
optim.
method
150
100
SSQ   Yi  W (ti ) 
2
50
0
-50
0
10
20
30
Time (days)
40
50
60
Optimise
 Many packages provide standard error of estimate (s.e.)
 Confidence: best value ±1.96 s.e. (or use t-distribution)
– assumes sampling distribution of parameter is normal (or Stud.-t)
350
300
Body weight (µg)
250
200
150
100
50
Wm = 294 µg
rB = 0.0690 day-1
0
-50
0
10
20
30
Time (days)
40
(241 - 347)
(0.0498 - 0.0881)
50
60
Intervals on the curve
 Reliability of the model curve (confidence interval)
 Where to expect new observations (prediction interval)
350
300
Body weight (µg)
250
200
150
100
50
0
-50
0
10
20
30
Time (days)
40
50
60
Remarks …
 Parameter estimates and intervals follow from
assumptions (the ‘error model’):
– model is correct
in this case: W0 is known
x-values are known (controlled)
errors in y are normal
errors in y are independent
errors in y constant variance
many data points (asymptotic)
350
300
250
Body weight (mg)
–
–
–
–
–
–
200
150
100
50
0
-50
0
10
20
30
Time (days)
40
50
60
Summary curve fitting
What do you need?
 Data set and a model for the process (curve)
 ‘Error model’ for the deviations
– normal, independent, homogeneous variances:
• leads to least squares
– not normal, and/or not homogenous variances:
• transform (log-transforming is popular)
• use different error model (see: likelihood)
Results are meaningful only when both process and error
model are meaningful
Current practice 1
Analysis of survival data
Type of endpoints
Mortality/Immobility = Quantal
count number of animals responding
– e.g., 8 out of 20 (is 40%)
– always whole number (or 0-100%)
– e.g., LC50 (conc. at which 50% of population is dead)
Growth/Reproduction = Graded
measure degree of response for each individual
– e.g., 85 eggs or body weight of 23.2 mg
– between 0 and infinite
– e.g., EC50 (conc. at which the mean response is 50%)
Survival analysis
Typical data set
– number of live (mobile) animals: quantal data
– example: Daphnia exposed to nonylphenol
mg/L
0h
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 percentage survival after 48 h
– concentration on log scale
Objective
100
survival (%)
– derive LC50
80
60
40
20
0
0.001
0.01
0.1
concentration (mg/L)
1
What model?
Requirements curve
– start at 100% and monotonically decreasing to zero
– inverse cumulative distribution?
– choose log-normal
survival (%)
100
80
60
40
20
0
0.001
0.01
0.1
concentration (mg/L)
1
Graphical method
Probit transformation
std. normal distribution + 5
mortality (%)
100
80
60
40
20
data
0
0.001
0.01
0.1
concentration (mg/L)
1
2 3 4 5 6 7 8 9
probits
Linear regression on probits versus log concentration
Fit model, least squares?
survival (%)
100
80
60
40
20
0
0.001
0.01
0.1
concentration (mg/L)
1
Fit model, ‘likelihood’
mg/L
0h
48 h
pi
0.004
20
20
≈1
0.032
20
20
≈1
0.056
20
20
≈1
0.100
20
20
≈1
0.180
20
16
≈0.8
0.320
20
2
≈0.1
0.560
20
0
≈0
 20  16
4
P(Y  16; p )    p (1  p )
 16 
Fit model, ‘likelihood’
mg/L
0h
48 h
pi
0.004
20
20
≈1
0.032
20
20
≈1
0.056
20
20
≈1
0.100
20
20
≈1
20
16
optim.
2
method
0
0.180
0.320
0.560
parameters
20
dose-resp.
20
model
b
p
a
c
≈0.8
≈0.1
data
≈0
‘error’
20

 16
4
predicted
model
P(Y  16;effect
p )    p (1  p )
 16 
Find a and b such
that probability of
data is maximised
Which model curve?
Popular distributions
– log-normal (‘probit’)
– log-logistic (‘logit’)
– Weibull
b
p
a
c
Which model curve?
LC50
log lik.
fraction surviving
1
0.9
Log-logistic
0.225
-16.681
0.8
Log-normal
0.226
-16.541
0.7
Weibull
0.242
-16.876
0.6
Gamma
0.230
-16.582
0.5
0.4
0.3
0.2
0.1
0
data
log-logistic
log-normal
Weibull
gamma
-1
10
concentration
Non-parametric analysis
Spearman-Kärber: wted. average of midpoints
survival (%)
100
 weights: number of
deaths in interval
 for symmetric
distribution (on log
scale)
80
60
40
20
0
0.001
0.01
0.1
log concentration (mg/L)
1
‘Trimmed’ Spearman-Kärber
100
survival (%)
Interpolate at 95%
80
60
40
20
0
0.001
Interpolate at 5%
0.01
0.1
log concentration (mg/L)
1
Distribution … of what?
Perhaps ‘tolerance’ … let’s assume:
probability density
– animal dies instantly when exposure exceeds ‘threshold’
– threshold varies between individuals
concentration
Concept of ‘tolerance’
1-cumulative density
1
80
60
20% mortality
40
20
0
0.001
0.01
0.1
1
concentration (mg/L)
probability density
survival (%)
100
20% mortality
What is the LC50?
1-cumulative density
1
80
60
?
40
20
0
0.001
50% mortality
0.01
0.1
1
concentration (mg/L)
probability density
survival (%)
100
50% mortality
Summary: survival data
Survival data are ‘quantal’ responses
– data are fraction of individuals responding
– one possible mechanism is ‘tolerance distribution’
Analysis types
– regression (e.g., log-logistic or log-normal)  LC50 or LCx
– non-parametric (e.g., Spearman-Kärber)  LC50
LC50 is …
– estimated concentration at which 50% of the population is dead
– (for specific exposure time/test conditions)
Current practice 2
Analysis of continuous data
Type of endpoints
Mortality/Immobility = Quantal
count number of animals responding
– e.g., 8 out of 20 (is 40%)
– always whole number (or 0-100%)
– e.g., LC50 (conc. at which 50% of population is dead)
Growth/Reproduction = Graded
measure degree of response for each individual
– e.g., 85 eggs or body weight of 23.2 mg
– between 0 and infinite
– e.g., EC50 (conc. at which the mean response is 50%)
Analysis of continuous data
Endpoints for individual
– in ecotoxicology, usually growth or reproduction
Two approaches
– hypothesis testing: NOEC/LOEC
– curve fitting: EC50 or in general: ECx
Regression modelling
Select model
– log-logistic most popular
– we cannot talk about a tolerance distribution!
120
total offspring/female
survival
20
15
LC50
10
5
0
100
80
EC50
60
40
20
0
1
10
concentration (mg/L)
100
0
0.1
1
concentration (mg/L)
10
Least-squares estimation
reproduction (#eggs)
100
80
60
40
20
0
0.001
0.01
0.1
concentration (mg/L)
1
Example: Daphnia repro
Plot concentration on log-scale
100
# juv./female
90
80
70
60
50
40
30
20
10
0
-2
10
-1
10
0
10
concentration
1
10
Example: Daphnia repro
Fit sigmoid curve
 Estimate ECx from the curve
100
EC10
0.13 mM
(0.077-0.19)
# juv./female
90
80
70
60
EC50
0.41 mM
(0.33-0.49)
50
40
30
20
10
0
-2
10
-1
10
0
10
concentration
1
10
Summary: continuous data
Repro/growth data are ‘graded’ responses
– look at mean response of individual animals
– not fraction of animals responding
– thus, we cannot talk about ‘tolerance distribution’
Analysis types
100
EC10
0.13 mM
(0.077-0.19)
90
80
# juv./female
– statistical testing  NOEC
– regression (e.g., log-logistic)  ECx
– note: estimates and intervals assume …
70
60
EC50
0.41 mM
(0.33-0.49)
50
40
30
20
10
0
-2
10
-1
10
0
10
concentration
1
10
Critical notes
Limitations of current practice
If EC50 is the answer …
… what was the question?
total offspring
“What is the concentration of chemical X that leads to 50%
effect on the total number of offspring of Daphnia magna
(Straus) after 21-day constant exposure under
standardised laboratory conditions?”
EC50
log concentration
Why 21 days?
Toxicity is a process in time …
 statistics like LC50/ECx/NOEC change in time
 this is hidden by strict standardisation
–
–
–
–
–
Daphnia mortality:
fish mortality:
Daphnia repro
fish growth
…
2 days
4 days
21 days
28 days
Survival effects
1
LC50
s.d.
tolerance
24 hours
0.370
0.306
48 hours
0.226
0.267
0.9
fraction surviving
0.8
0.7
0.6
0.5
24 hours
0.4
0.3
48 hours
0.2
0.1
0
0
0.1
0.2
0.3
0.4
concentration
0.5
0.6
0.7
Sub-lethal tests
With time, control response increases and all
parameters may change …
100
increasing time (t = 9-21d)
# juv./female
90
80
70
60
50
40
30
20
10
0
-2
10
-1
10
0
10
concentration
1
10
EC10 in time
survival
Alda Álvarez et al. (2006)
body length
cumul. reproduction
carbendazim
2.5
pentachlorobenzene
140
120
2
100
1.5
80
60
1
40
0.5
20
0
0
5
10
time (days)
15
20
0
0
2
4
6
8
10
time (days)
12
14
16
Concluding on test duration
Effects change in time, how depends on:
– endpoint, species, chemical …
– environmental conditions
No such thing as the ECx
– always report endpoint and test duration
– is it a useful summary statistic?
Watch out when …
– comparing species, chemicals, endpoints …
– using ECx to judge ‘safety’ of a chemical …
What about alternatives?
 Rethink the question leading to ECx …
“What is the concentration of chemical X that leads to 50%
effect on the total number of offspring of Daphnia magna
(Straus) after 21-day constant exposure under
standardised laboratory conditions?”
 Better questions start with “How …” or “Why …”
 But, biology is so complex …
Dealing with complexity
Environmental chemistry …
– predict the concentrations of chemicals in the environment
– from emissions and physico-chemical properties
Make an idealisation
 E.g., multimedia-fate or ‘box’ models
air
water
natural
soil
agricult. industr.
soil
soil
sediment
emission
advection
diffusion
degradation
For effects: TKTD modelling
toxicodynamics
external
concentration
(in time)
toxico-kinetic
model
internal
concentration
in time
process model
for the organism
toxicokinetics
effects on
endpoints
in time
More information
on TKTD modelling: www.debtox.info
Summercourse dynamic modelling for toxic
effects, August 2018 (DK)
Mass & energy conservation
Mass & energy conservation
Mass & energy conservation
Mass & energy conservation
Mass & energy conservation
Dynamic Energy Budget
Organisms obey mass and energy conservation
– find the simplest set of rules ...
– over the entire life cycle ...
– for all organisms (related species follow related rules)
offspring
growth
maturation
maintenance
DEBtox basics
Assumptions
- effect depends on internal concentration
- chemical changes parameter in DEB model
DEB parameter
toxicokinetics
NEC
DEB
blank value
internal concentration
growth and repro in time
body length
cumulative offspring
Ex.1: maintenance costs
time
Jager et al. (2004)
TPT
time
body length
cumulative offspring
Ex.2: growth costs
time
Alda Álvarez et al. (2006)
Pentachlorobenzene
time
Ex.3: egg costs
body length
cumulative offspring
Chlorpyrifos
time
Jager et al. (2007)
time