Uncertainty Analysis of
Dose-Response Data with
Threshold Modeling
Jeff Swartout, U.S. EPA, ORD, NCEA, Cincinnati, OH
Office of Research and Development
Full Name of Lab, Center, Office, Division or Staff goes here. <Go to View, Master, Title Master to change>
October 30, 2007
Purpose and Motivation
• Provide alternative to a non-zero BMR
• Consistency in risk estimates
• Compare threshold vs. non-threshold approaches
1
Population Threshold Concept
• Considering only adverse (toxic) effects, such as
functional damage to an organ system or death, in the
extreme, there must be some level of exposure, below
which the effect does not occur in any individual (Cox,
1987).
• One molecule may destroy an enzyme or disrupt a
membrane but cannot, by itself, result in functional
damage unless the effect is fixed and heritable.
2
Rat data
HED extrapolation
Non-threshold fit
Threshold fit
RfD
+
0.0010
0.0100
+
0.0001
cumulative response
0.1000
1.0000
Threshold vs. Non-Threshold
0.1
1.0
10.0
dose
3
100.0
Threshold Dose-Response Models
• Individual tolerance distributions
– Lognormal, Weibull, log-logistic, etc.
– No population threshold
• Population threshold models
– Tolerance distribution with D - T term
– Tukey-lambda family (Cox, 1987)
4
Threshold Models
Hill (log-logistic)
(D − T )
N
N
ED50 + ( D − T )
N
Pareto
⎛T ⎞
1− ⎜ ⎟
⎝α ⎠
−α
Weibull
⎡ ⎛ D − T ⎞c ⎤
1 − exp ⎢ − ⎜
⎟ ⎥
⎢ ⎝ b ⎠ ⎥
⎦
⎣
5
D = administered dose
T = threshold dose parameter
N = Hill exponent
C = Weibull power
Bootstrap Procedure
• Fit threshold models to raw data
• Select best-fitting model
• Compute “true” response for each dose
– Use non-threshold model fit for zero-response dose
groups
6
Bootstrap Procedure
• Generate random binomial response for each dose
7
group (parametric bootstrap)
– Simulates re-running the experiment at fixed doses
with random draws from the same population, given
the true probability of response at each dose = pd
– Generates a new response vector (number of
responders)
• rbinom(nd, pd)
– nd is the number of individuals in dose group d
– pd is fitted response to raw data
• Fit all models to bootstrapped response
• Save threshold estimates at each iteration from bestfitting model
Assumptions and Limitations
• True animal response represented by initial model fit
• Response at zero-observed response doses equivalent
to fitted non-threshold response (divided by 2)
• Assumed response distribution valid near threshold
• Binomial uncertainty only
• Constraints on parameter space are ignored
8
Sample Bootstrap Output
(Frambozadrine)
0.050
0.001
0.005
cumulative response
0.500
Hill
Weibull
gamma
lognormal
Pareto
1
5
10
BMDL10
50
100
dose (mg/kg-day)
Threshold fits shown in relation to the BMDL
9
Colors indicate best-fitting model at each iteration (100 shown)
0.10
0.05
cumulative response
0.50
1.00
Sample Bootstrap Output
(Mordorine)
0.001
0.010
BMDL10
0.100
1.000
10.000
dose
Threshold fits shown in relation to the BMDL
10
Colors indicate best-fitting model at each iteration (100 shown)
0
0.0
1
0.5
2
1.0
3
4
1.5
5
2.0
6
Sample Bootstrap Threshold
Distributions
0.5
1.0
1.5
Threshold (log10 mg/kg-day)
2.0
-2.0
-1.5
-1.0
-0.5
0.0
Threshold (log10 mg/kg-day)
0.5
1.0
0.0
0.0
0.2
0.2
0.4
0.4
0.6
0.6
0.8
0.8
1.0
1.0
1.2
0.0
-4
11
-3
-2
-1
Threshold (log10 mg/kg-day)
0
1
-3
-2
-1
0
Threshold (log10 mg/kg-day)
1
1.0
Frambozadrine
0.6
0.4
0.0
0.2
cumulative response
0.8
Weibull (BMD)
Weibull threhsold
BMDL---BMD
T05---Tml
5
10
50
dose (mg/kg-day)
12
100
1.0
Frobozinate
0.6
0.4
0.0
0.2
cumulative response
0.8
log-logistic (BMD)
Pareto
BMDL---BMD
T05---Tml
0.1
0.5
1.0
5.0
dose (mg/kg-day)
13
10.0
50.0
100.0
1.0
Gruesite
0.6
0.4
0.0
0.2
cumulative response
0.8
log-logistic (BMD)
Pareto
BMDL---BMD
T05---Tml
10^-4
10^-3
10^-2
10^-1
10^0
dose (mg/kg-day)
14
10^1
10^2
10^3
0.4
0.6
log-normal (BMD)
log-normal threhsold
BMDL---BMD
T05---Tml
0.0
0.2
cumulative response
0.8
1.0
Phluginium
0.5
1.0
5.0
dose (mg/kg-day)
15
10.0
50.0
0.4
0.6
Weibull (BMD)
Pareto
BMDL---BMD
T05---Tml
0.0
0.2
cumulative response
0.8
1.0
Mordorene
0.01
0.10
1.00
dose (mg/kg-day)
16
10.00
1.0
Neelixir
0.6
0.4
0.0
0.2
cumulative response
0.8
Weibull (BMD)
Weibull threhsold
BMDL---BMD
Tml
0.1
1.0
10.0
dose (mg/kg-day)
17
100.0
Summary of Results
Compound
Weibull
Power
BMDL10a
TLb
BMDLrc
TLrd
TL: BMDL
Frambozadrine
1.4
24.6
4.45
1.7
4.8
0.18
Frobozinate
0.34
0.10
0.091
3.0
11
0.91
Gruesite
0.35
5.7 x 10-5
0.27
2940
3.7
4780
Phluginium
0.83
1.89
0.19
1.2
5.1
0.10
Mordorene
0.86
0.0027
0.030
217
33
11
Neelixir
1.5
18.1
0
0.25
–
0.22e
a95%
lower confidence bound on BMD10 (BMDS)
b95% lower confidence bound on threshold (bootstrap)
cRatio of BMDMLE to BMDL
dRatio of TMLE to TL
eTMLE : BMDMLE
18
1.0
That’s All
0.500
Hill
Weibull
gamma
lognormal
Pareto
0.005
0.050
cumulative response
0.6
0.4
0.0
0.001
0.2
cumulative response
0.8
Weibull (BMD)
Weibull threhsold
BMDL---BMD
T05---Tml
1
5
10
50
5
100
3
2
0
1
Probability density
4
5
6
dose (mg/kg-day)
0.0
0.5
1.0
Threshold (log10 mg/kg-day)
19
10
dose (mg/kg-day)
1.5
2.0
50
100