reliability of analytical methods’ results: a bayesian ... · ml t i i i x p z x p z..,,. .,,...
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Reliability of Analytical Methods’Results: a Bayesian Approach to
Analytical Method Validation
E. Rozet1, B. Govaerts2, P. Lebrun1, B. Boulanger3, E.
Ziemons1, Ph. Hubert1
1 Laboratory of Analytical Chemistry, ULg, Liège, Belgium2 Institut de Statistique, UCL, Louvain-la-Neuve, Belgium.3 Arlenda SA, Liège, Belgium.
UNIVERSITE de LIEGE
Agrostat 2012 Paris, March 1, 2012
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Analytical Method Life Cycle
• What is the final aim of quantitative analytical
methods ?
– Start with the end !
– Objective: provide results used to make decisions
• Release of a batch
• Stability/Shelf life
• Patient health
• PK/PD studies, …
• What matters are the results produced by the
method.
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Analytical Method Life Cycle
Development
ValidationRoutine
Use
Selection
Life Cycle
Routine
use
Routine Use Validation
Guarantees ?
Reliability ?
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Analytical Method Life Cycle
• Need to demonstrate/guarantee that the
analytical method will provide, in its future
routine use, quality results
• This is the key aim of Analytical Method
Validation !
How ?
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Analytical Method Validation
• Traditional vision:
– The Validation Criteria Check List:
• Selectivity
• Trueness/Mean Accuracy
• Precision
• Linearity
• Range
• Limit of Quantification (LOQ)
Method Valid !
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Analytical Method Validation
• Traditional vision:
– Is a valid method providing reliable results ?
Bia
sP
recis
ion
% Bias< 3%
% CV< 2%
Analytical Results
Are you ready to takethis risk?
Analytical Method
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Aim of validationIs to give to laboratories as well as to regulatory agenciesthe guaranties that each result that will be obtained in routine will be close enough to the unknown true value ofthe analyte in the sample.
Analytical Method Validation
[ ]minπλπ ≥<−= Ti µXP
πmin
= minimum probability that a result will be included inside ± λ
λλλλ= predefined acceptance limits
λµ −T λµ +
Tµ
π
E. Rozet et al., J. Chromatogr.A, 1158 (2007) 126
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Series 1
Series 2
Series J
Validation standards
K repetitions
Typical Validation Design
Known concentration
Measu
red
co
nce
ntr
ati
on
Known concentration
Measu
red
co
nce
ntr
ati
on
Known concentration
Measu
red
co
nce
ntr
ati
on
I concentration levels
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Typical Statistical Model
• By concentration level i:
– One Way Random ANOVA model
– Intermediate Precision variance
( )( )2
,,
2
,,
,,,
,0~
,0~
ijki
iji
jkijiijki
N
N
X
ε
α
σε
σα
εαµ ++=
2
,
2
,
2
.,. iiiPI εα σσσ +=
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Reliability Probability Estimator 1 – πBetii
• Based on β-expectation toleranceintervals:
Allows to predict whereeach future result will fall(Wald, 1942).
λµ −T λµ +
Tµ
β
Acceptance Limits
Tolerance Interval
� If the ββββ-expectation tolerance interval is includedinside the acceptance limits, then the probability thateach future result will be within the acceptance limits
is at least ββββ (ex. 80%).
B. Boulanger et al., J. Chromatogr. B, 877 (2009) 2235
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Reliability Probability Estimator 1 – πBetii
• Based on β-expectation toleranceintervals:
Beti
iπ
λµ −T λµ +
Tµ
Acceptance Limits
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Reliability Probability Estimator 1 – πBetii
• Based on β-expectation toleranceintervals:
• N=JK.
• is the mean results
• t(f): Student distribution with f degrees of freedom using
Satterthwaite approximation
•
[ ] [ ]
( )( )
( )
( )( )
( )
+
++
−+<+
+
++
−−>=
+<+−>=
1ˆ
1ˆ1ˆ
1ˆ
1ˆ1ˆ
.,.
,
.,.
,
,,
i
i
iPI
iiT
i
i
iPI
iiT
iTiiTi
Beti
i
RN
RK
XftP
RN
RK
XftP
XPXP
σ
λµ
σ
λµ
λµλµπ
iX
2
2
ˆ
ˆˆ
ε
α
σ
σ=iR
W. Dewé et al., Chemometr. Intell. Lab. Syst. 85 (2007) 262-268.
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Reliability Probability Estimator 2 – πMLi
• Maximum likelihood estimator
where Z is a standard normal variable.
( ) ( )
−+<+
−−>=
iPI
iiT
iPI
iiTML
i
XZP
XZP
.,.
,
.,.
,
ˆˆ σ
λµ
σ
λµπ
B. Govaerts et al., Qual. Reliab. Engng. Int. 24 (2008) 667-680.
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Bayesian Reliability Estimator - π
=
j
j
j u
u
,1
,0U are the random effects of the j
th runs
• Aims: modeling the reliability probability over the whole concentration range
• Model: Linear model with random slopes and intercepts
ijkiTjjiTijk uuX εµµββ ++++=,,1,0,10
θ
=
1
0
β
β are the fixed effects
jU ~ ),(22
2
xuiN Σ0 σ
( )2,0~ iijk N σε
θ ~
Γ,
1
0N 0Γ =−1
Σ ~ )2,0001.0(Wishart 2I
γ ~ )0001.0,0(N
στ
1= ~ )0001.0,0001.0(Gamma
( )γµσσ iTi ,=
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Simulations
• 4 scenarios:
– Conditions
• Analytical Method relative bias: 0% and 10%
• Analytical Method I.P. RSD: 6.5% and 16%
• Known concentrations (µT,i):60%, 80%, 100% and 120%
• Acceptance limits: λ=±20%
• Nb Series: J=4
• Nb Repetitions: K=4
– Criteria
• Compare median estimated reliability probabilities to true
probability
• Compare ranges (min to max) of estimated reliability
probabilites
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Case 1: 0% bias – 16.0% RSD
Median values
True ππππ
ππππML
ππππ
ππππBetiππππππππMLππππBeti
Ranges
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Case 2: 10% bias – 16.0% RSD
ππππππππML
ππππBeti
Ranges
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Case 3: 0% bias – 6.5% RSD
ππππ
ππππML
ππππBeti
Ranges
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Case 4: 10% bias – 16.0% RSD
ππππππππML
ππππBeti
Ranges
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Example of application
• Validation of a bioanalytical method:
– SPE-HPLC-UV method for the quantification ofketoglutaric acid (KG) and hydroxymethylfurfural(HMF) in human plasma
• Known concentrations (µT,i): 0.13, 0.67, 3.33, 66.67 and 133.33 µg/ml
• Nb Series: J=3
• Nb Repetitions: K=4
• Acceptance limits: λ=±20%
• Minimum reliability probability: πmin=0.90
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Ketoglutaric acid
LOQ
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Hydroxymethylfurfural
LOQ
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Conclusions
� Switch from the traditional check list validation to a rewarding, useful and predictive method validation
� The quality of future results (π) must be the objective of method validation and not the past performances of the method.
� The Bayesian reliability probability estimator is less biasedand more precise.
� In such a way, the risks are known at the end of the validation.
� This decision methodology is fully compliant with actual regulatory requirements
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Thanks for your attention
• Check our publications at:
http://orbi.ulg.ac.be/
• Contact:
UNIVERSITY of LIEGE