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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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3

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 ,=

8

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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

10

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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:

Eric.Rozet@ulg.ac.be

UNIVERSITY of LIEGE