Kinetic Models Considered
Jeremy Dyson
Basel, Switzerland
Introduction
Selected core models for parent compounds &
metabolites in soil and water-sediment studies
Models can be applied to other studies, e.g. hydrolysis
Models chosen as simple & sensible approaches to
representing behaviour from pseudo-mechanistic to
more pragmatic perspectives
Models displayed in terms of:
 Integrated equations based on initial conditions
 Differential of above for solving new problems
 Differential equations without initial conditions
 Endpoint calculation (DT50/DT90) from parameters
2
Outline
The Core Models
 Single First Order
 Biphasic Models
– Gustafson & Holden
– Bi-Exponential
– Hockey Stick

Lag Phase Models
– Modified Hockey Stick
– Logistic
Alternatives & Links to Leaching Models
Conclusion
3
Single First Order (SFO)
% Remaining
Single First Order
100
90
80
70
60
Fixed Shape
DT90 = 3.32 x DT50
Assumes microbes not limiting
Degradation  % Remaining
DT50
50
40
30
20
10
0
DT90
0
1
2
3
Time / DT50
4
4
5
Single First Order (SFO)
Equation (integrated form)
M  M0 e k t
Underlying differential equation
dM
 k M
dt
where
M = Total amount of chemical present at time t
M0 = Total amount of chemical present at time t=0
k = Rate constant
Parameters to be determined
M0, k
Endpoints
100
DTx  100  x
k
ln 2
DT50 
k
ln10
DT90 
k
ln
5
Biphasic Models
% Remaining
SFO vs. Biphasic
100
90
80
70
60
DT50
50
40
30
20
10
0
0
Generally:
DT50s Shorter
DT90s Longer
DT90
1
2
3
Time
6
4
5
Biphasic Models
Some Causes of Biphasic Behaviour

Aged sorption
Non-linear sorption
Declining microbial activity
Spatial variations in field

Seasonal changes in weather



7
Gustafson & Holden (FOMC)
Equation (integrated form)
M
M0
t

  1
 

Differential equation
(to be used only for parameter estimation)
1
dM

 M
dt

t

  1



where
M = Total amount of chemical present at time t
M0 = Total amount of chemical applied at time t=0


= Shape parameter determined by coefficient of variation of k values
= Location parameter
Parameters to be determined
M0, , 
Endpoints
 1


 
 100     

DTx   
1
 100  x 



  1  
DT50    2     1




  1  
DT90   10     1




8
Note: The proposed equation differs slightly from that in the original Gustafson and Holden (1990)
reference. The parameter  corresponds to 1 /  in the original equation.
Bi-Exponential (DFOP or FOTC)
Equation (integrated form)
Differential equation
(to be used only for parameter estimation)
M  M1 e k 1 t  M2 e k 2 t
or

M  M0 g e k 1 t  1  g e k 2 t
where
M =
M1 =
M2 =
M0 =
g
=
k1 =
k2 =

dM
k g e k 1 t  k 2 1  g e k 2 t
 1
M
dt
g e k 1 t  1  g e k 2 t
Total amount of chemical present at time t
Amount of chemical applied to compartment 1 at time t=0
Amount of chemical applied to compartment 2 at time t=0
M1 + M2 = Total amount of chemical applied at time t=0
fraction of M0 applied to compartment 1
Rate constant in compartment 1
Rate constant in compartment 2
Parameters to be determined
M1, M2, k1, k2 or M0, g, k1, k2
Endpoints
An analytical solution does not exist.
DTx values can only be found by an iterative procedure
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Hockey Stick (HS)
Equation (integrated form)
Underlying differential equation
M  M0 e k1 t
for ttb
dM
 k1 M
dt
for ttb
M  M0 e k1 tb e k 2 t  tb 
for t>tb
dM
 k 2M
dt
for t>tb
where
M = Total amount of chemical present at time t
M0 = Total amount of chemical applied at time t=0
k1 = Rate constant until t=tb
k2 = Rate constant from t=tb
tb = Breakpoint (time at which rate constant changes)
Parameters to be determined
M0, k1, k2, tb
Endpoints
DTx 
ln
100
100  x
k1
100


ln 100  x  k1tb 

DTx  tb  
k2
10
if DTxtb
if DTx>tb
Lag Phase Models
SFO vs. Lag Phase
% Remaining
Lag Phase
100
90
80
70
60
50
40
30
20
10
0
0
1
2
3
Time
11
4
5
Modified Hockey Stick
Equation (integrated form)
M  M0
for ttb
M  M0 e k t  tb 
for t>tb
Underlying differential equation
dM
0
dt
dM
 k M
dt
where
M = Total amount of chemical present at time t
M0 = Total amount of chemical applied at time t=0
k = Rate constant from t=tb
tb = Breakpoint (time at which decline starts)
Parameters to be determined
M0, k, tb
Endpoints
DTx 
12
ln
100
100  x
k
or DTx 
ln
100
100  x  t
b
k
for ttb
for t>tb
Logistic
Equation (integrated form)
M  M0 [
amax
amax  a0  a0 e(r t )
Differential equation
(to be used only for parameter estimation)
a max
] r
a
where
M =
M0 =
amax =
a0 =
r
=
a0 amax
a0  (amax  a0 ) e( r t )
Total amount of chemical present at time t
Total amount of chemical applied at time t = 0
Maximum value of degradation constant (reflecting microbial activity)
Initial value of degradation constant
Microbial growth rate
Note:
For a0 =amax (i.e. activity of degrading microorganisms is already at its maximum at
the start of the experiment) the model reduces to SFO kinetics with rate constant amax
Parameters to be determined:
M0, amax, a0, r
Endpoints
a
1
100
DTx  ln [1  max (1 
r
a0
100  x
r / amax
a
1
DT50  ln [1  max (1  2r / amax )]
r
a0
a
1
DT90  ln [1  max (1  10r / amax )]
r
a0
13
)]
Alternatives & Links to Leaching Models
Alternatives models can be used, but:


Needs to be justified, e.g. Michaelis-Menten kinetics
Need to avoid, where possible:
– Time-dependent endpoints
– Concentration-dependent endpoints
– Large number of parameters
– Microbial population dynamics

Avoid Timme et al. as already noted in Guideance
Identify biphasic kinetics for use in leaching models
 Not possible, but can link to DFOP for estimation
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Alternatives & Links to Leaching Models
Equilibrium
Sorption
Kinetic
Sorption
15
Non-Degrading
Degrading
Compartment
Compartment
Conclusion
Core models able to handle most kinetics problems
Hence focus on solving problems using these models
16