Dirk Young (USEPA)

Ian Kennedy (Health Canada)
Problem: Find a First-Order Degradation 

Representation for Non-First Order Data

0

0.2

0.4

0.6

0.8

1

1.2

0 50 100 150 200 250 300 350 400

Concentration (C/C

0

)

Days

Data

K=0.01

K= 0.02

K= 0.005

C = C0e

-kt
Right Answer is Dependent on Desired 

Protectiveness

0

0.2

0.4

0.6

0.8

1

1.2

0 50 100 150 200 250 300 350 400

Days

Concentration
Approaches

 FOCUS: 

 Use several models: SFO, FOMC, Hockey Stick

 Find best fit model per chi square method

 Derive a DT90 from the best fit model

 Use slower DFOP or HS rate as a lower bound

 NAFTA:

 Determine whether data are significantly non-first order

 If not first order, transform IORE parameters to first order via DT90

 Use slower DFOP rate as lower bound
Questions

 What statistics do you use to select the “best” model? 

 Is a precise estimate of DT90 necessary?

 What about extrapolation (data do not reach DT90)? 
“Perfect” estimates of DT90 do not assure consistent assessments.

NAFTA emphasis on developing a consistent procedure.

The basis for DT90 selection was qualitative. So does 

precise estimation offer anything?

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 12

Concentration

Time

First Order Model

Data Set #1

Data Set #2

DT90
“Extrapolation” --not really. How about 

Rate Transformation?

k N

k

N

iore 1

1 0.10 1

ln 0.10

1

0

0.2

0.4

0.6

0.8

1

1.2

0 50 100 150 200 250 300

Days

Concentration

Transforming IORE parameters to 1st order parameter

Acceptance criteria should not be whether data reaches DT90, but rather if 

there are sufficient data for confident model parameters.

(As dictated by acceptance criteria of study?).
NAFTA simplification

 Use a single model for DT90 determination

 Eliminate statistics for determining best model

 Eliminate extrapolation criteria
Step 1. Prepare Data

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5

Concentration

Time
Step 2. Fit SFO and IORE

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6

Concentration

Time

First Order Fit

IORE Fit

Data

N

kC

dt

dc IORE:

SFO when N=1,
Step 3: Make Determination of FO or not

0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6

Concentration

Time

First Order Fit

IORE Fit

Data

k

N

Sc

1

2

3

4

First Order Fit (N=1)

50% Confidence Region

N

kC

dt

dc

When N=1, First Order Model
Step 4: Calculate a representative half life from IORE

trep

0

20

40

60

80

100

120

0 50 100 150 200

Concentration

Time

IORE Representative (IORE) Data
Step 5: Estimate DFOP Parameters (Co, k1, k2, g)

0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6

Concentration

Time

DFOP Data
Step 6: Compare IORE Representation to DFOP Slow Rate: 

Choose the Lower of the Two

0

20

40

60

80

100

120

0 50 100 150 200

Concentration

Time

Representative (IORE)

DFOP fit

DFOP slow rate

Data
Use SFO

Use IORE t

rep

Use DFOP 

slow half life

no

yes no

yes

Summary

Is SFO fit 

acceptable? 

Is IORE t

rep< DFOP 

slow half life?
Status

 NAFTA Project Completed & Signed Nov 2011

 Working out technical details regarding tool: 

 Evaluating an R program created by PMRA

 Excel versions less automatic

 Implementation plan in works. 

 Re-evaulation after 1 year.