# Modelling uncertainty estimation for the determination of aflatoxin M(1) in milk by visual and densitometric thin-layer chromatography with immunoaffinity column clean-up.

**ABSTRACT** The uncertainty of aflatoxin M(1) concentration in milk, determined by thin-layer chromatography (TLC) with visual and densitometric quantification of the fluorescence intensities of the spots, was estimated using the cause-and-effect approach proposed by ISO GUM (Guide to the expression of uncertainty in measurement) following its main four steps. The sources of uncertainties due to volume measurements, visual and densitometric TLC calibration curve, allowed range for recovery variation and intermediary precision to be taken into account in the uncertainty budget. For volume measurements the sources of uncertainties due to calibration, resolution, laboratory temperature variation and repeatability were considered. For the quantification by visual readings of the intensity of the aflatoxin M(1) in the TLC the uncertainty arising from resolution calibration curves was modelled based on the intervals of concentrations between pairs of the calibration standard solutions. The uncertainty of the densitometric TLC quantification arising from the calibration curve was obtained by weighted least square (WLS) regression. Finally, the repeatability uncertainty of the densitometric peak areas or of the visual readings for the test sample solutions was considered. For the test samples with aflatoxin M(1) concentration between 0.02 and 0.5 µg l(-1), the relative expanded uncertainties, with approximately 95% of coverage probability, obtained for visual TLC readings were between 60% and 130% of the values predicted by the Horwitz model. For the densitometric TLC determination they were about 20% lower. The main sources of uncertainties in both visual and densitometric TLC quantification were the intermediary precision, calibration curve and recovery. The main source of uncertainty in the calibration curve in the visual TLC analysis was due to the resolution of the visual readings, whereas in the densitometric analysis it was due to the peak areas of test sample solutions followed by the intercept and slope uncertainties of the calibration line.

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**ABSTRACT:**A collaborative study was conducted to evaluate the effectiveness of an immunoaffinity column cleanup liquid chromatographic method for determination of aflatoxin M1 in milk at proposed European regulatory limits. The test portion of liquid milk was centrifuged, filtered, and applied to an immunoaffinity column. The column was washed with water, and aflatoxin was eluted with pure acetonitrile. Aflatoxin M1 was separated by reversed-phase liquid chromatography (LC) with fluorescence detection. Frozen liquid milk samples both naturally contaminated with aflatoxin M1 and blank samples for spiking, were sent to 12 collaborators in 12 different European countries. Test portions of samples were spiked at 0.05 ng aflatoxin M1 per mL. After removal of 2 noncompliant sets of results, the mean recovery of aflatoxin M1 was 74%. Based on results for spiked samples (blind pairs at 1 level) and naturally contaminated samples (blind pairs at 3 levels) the relative standard deviation for repeatability (RSDr) ranged from 8 to 18%. The relative standard deviation for reproducibility (RSDR) ranged from 21 to 31%. The method showed acceptable within- and between-laboratory precision data for liquid milk, as evidenced by HORRAT values at the low level of aflatoxin M1 contamination.Journal of AOAC International 84(2):437-43. · 1.23 Impact Factor - [show abstract] [hide abstract]

**ABSTRACT:**In the interlaboratory study programme "ILS Coal Characterisation", eight interlaboratory studies were organised based on the ISO standards for coal analysis. The use of blind samples in each round allows comparability of measurement results between rounds to be assessed. Based on the results, it could be demonstrated that the vast majority of the measurement results of the laboratories were traceable to results obtained in previous rounds of this programme. The hypothesis has been formulated that the combined standard uncertainty obtained from an interlaboratory study is equal to the reproducibility standard deviation. Whether the reproducibility can be used as the basis for the certification depends on whether the interlaboratory study includes all effects to be taken into account for establishing an uncertainty statement.Accreditation and Quality Assurance 10/1998; 3(11):462-467. · 1.13 Impact Factor - [show abstract] [hide abstract]

**ABSTRACT:**The possibility of using interlaboratory study repeatability and reproducibility estimates as the basis for measurement uncertainty estimates is discussed. It is argued that collaborative trial reproducibility is an appropriate basis for estimating uncertainty in routine testing provided certain conditions are met by the laboratory. The primary shortcomings relate to establishment of traceability and consequent estimation of bias associated with the method, and quantitatively establishing the relevance to the single laboratory. Approaches to resolving both difficulties are proposed, the former via full implementation of trueness determination suggested in ISO 5725 : 1994 or by independent checks on individual accuracy and precision, the latter via a reconciliation procedure. The paper also discusses other factors including sampling and sample pre-treatment, change in sample matrix, and the influence of level of analyte.Accreditation and Quality Assurance 02/1998; 3(3):95-100. · 1.13 Impact Factor

Page 1

Food Additives and Contaminants

Vol. 29, No. 4, April 2012, 679–693

Modelling uncertainty estimation for the determination of aflatoxin M1in milk by visual and

densitometric thin-layer chromatography with immunoaffinity column clean-up

K.L. Carvalhoa, G.A.A. Gonc ¸ alvesa, A.L. Lopesa, E.A. Santosa, E.A. Vargasaand W.F. Magalha ˜ esb*

aLaboratory of Quality Control and Safety Food – LACQSA/LANAGRO-MG/MAPA, Av. Raja Gabaglia, 245, Cidade Jardim,

CEP 30380-090 – Belo Horizonte, MG, Brazil;bChemistry Department, Universidade Federal de Minas Gerais – UFMG, Av.

Pres. Antonio Carlos, 6627, Campus Pampulha, CEP 31270-901 – Belo Horizonte, MG, Brazil

(Received 26 November 2010; final version received 7 November 2011)

The uncertainty of aflatoxin M1concentration in milk, determined by thin-layer chromatography (TLC) with

visual and densitometric quantification of the fluorescence intensities of the spots, was estimated using the cause-

and-effect approach proposed by ISO GUM (Guide to the expression of uncertainty in measurement) following

its main four steps. The sources of uncertainties due to volume measurements, visual and densitometric TLC

calibration curve, allowed range for recovery variation and intermediary precision to be taken into account in the

uncertainty budget. For volume measurements the sources of uncertainties due to calibration, resolution,

laboratory temperature variation and repeatability were considered. For the quantification by visual readings of

the intensity of the aflatoxin M1in the TLC the uncertainty arising from resolution calibration curves was

modelled based on the intervals of concentrations between pairs of the calibration standard solutions. The

uncertainty of the densitometric TLC quantification arising from the calibration curve was obtained by weighted

least square (WLS) regression. Finally, the repeatability uncertainty of the densitometric peak areas or of the

visual readings for the test sample solutions was considered. For the test samples with aflatoxin M1concentration

between 0.02 and 0.5mgl?1, the relative expanded uncertainties, with approximately 95% of coverage probability,

obtained for visual TLC readings were between 60% and 130% of the values predicted by the Horwitz model.

For the densitometric TLC determination they were about 20% lower. The main sources of uncertainties in both

visual and densitometric TLC quantification were the intermediary precision, calibration curve and recovery. The

main source of uncertainty in the calibration curve in the visual TLC analysis was due to the resolution of the

visual readings, whereas in the densitometric analysis it was due to the peak areas of test sample solutions

followed by the intercept and slope uncertainties of the calibration line.

Keywords: chromatographic analysis; clean-up – affinity columns; statistical analysis; regression; measurement

uncertainty; aflatoxins; mycotoxins – aflatoxins; mycotoxins; milk

Introduction

One of the most important metrological characteris-

tics of a measurement result is its uncertainty. It is a

worldwide consensus that a result is not complete

without an expression of its uncertainty (ABNT,

INMETRO 2003; BIPM et al. 1995) and its estima-

tion is a requirement for testing laboratories accred-

itationby theInternational

Standardization (ISO) (2005). The present paper

uses the methodology called the cause-and-effect

approach(Ellison and

Barwick and Ellison 1998), or GUM approach

(ABNT, INMETRO 2003; BIPM et al. 1995) or

bottom-up approach to estimate the uncertainty of

aflatoxin M1concentration in bovine milk which was

Organization for

Barwick1998a, 1998b;

determined by thin-layer chromatography (TLC) with

visual and densitometric quantification. Within this

methodology, this work addresses especially to the

following issues:

. Evaluation of the uncertainty due to the

resolutionof the

M1 in the TLC quantification by a visual

reading.

. Accounting for the heteroskedasticity of the

instrumental response (fluorescence intensity)

in the densitometric TLC calibration by using

weighted least squares (WLS).

. Inclusion of an uncertainty contribution due

to the uncorrected bias within an allowed

range for the recovery variation.

intensity ofaflatoxin

*Corresponding author. Email: welmag@ufmg.br

ISSN 1944–0049 print/ISSN 1944–0057 online

? 2012 Taylor & Francis

http://dx.doi.org/10.1080/19440049.2011.648959

http://www.tandfonline.com

Page 2

. Accounting for a within-laboratory reproduc-

ibility or intermediary precision through the

data of the analysis of control samples for

quality control.

Other approaches that estimate the uncertainty of

an analytical measurement make use of data obtained

from a method validation study (Ellison et al. 2000;

Barwick and Ellison 2000a, 2000b; Barwick et al. 2000)

or from interlaboratory studies through the results of

collaborative trials (Adriaan et al. 1998; Barwick and

Ellison 1998; Ellison 1998).

Recent publications have addressed considerations

to assess the uncertainty of aflatoxin M1determination

(Calaresu et al. 2006; Populaire and Gime ´ nez 2006).

Calaresu et al. (2006) used data obtained from preci-

sion, trueness and ruggedness studies during method

validationtoestimate

However, the uncertainties due to the input quantities

appearing in their measurand equation and in their

cause-and-effect diagram were not included in their

uncertainty budget. For instance, glassware tolerances

were used to estimate uncertainties and not the data

from the calibration certificates and laboratory tem-

perature variation which appear in the cause-and-effect

diagram. A very simple measurand equation was

presented where only three input quantities appear:

the volume of the sample taken for analysis, the final

volume of the sample solution, and the concentration

of aflatoxin evaluated from the calibration curve.

However, nothing was presented about the least square

procedure used to fit the calibration curve data, nor

about both the calibration curve range or the values of

the intercept, the slope of the calibration curve, and

their uncertainties and covariance.

Populaire and Gime ´ nez (2006) compared the

uncertainty estimation obtained from the bottom-up

method with the top-down method. They concluded

that the main sources of uncertainties in different

analytical methods were mainly due to the intermedi-

ary precision, accuracy/recovery and, in some cases,

calibration. They also concluded that the uncertainties

due to weighting and volume measurement were in

general negligible. However, no details of their calcu-

lations were given, also they used a poor calibration

design with only one standard calibration solution.

The present paper presents a detailed uncertainty

budget to estimate the measurement uncertainty for

aflatoxin M1determination in milk with immunoaffi-

nity column clean-up and thin-layer chromatographic

(TLC) determination with visual and densitometric

quantification.

Aflatoxin M1 (4-hydroxyaflatoxin B1, (6aR-cis)-

2,3,6a,9a-tetrahydro-9a-hydroxy-4-methoxycyclopenta

[c]furo[30,20:4,5]furo[2,3-h][l]benzopyran-1,11-dione) is

a genotoxic carcinogenic hydroxylated metabolite of

aflatoxin B1found in the milk of animals that have

uncertainty components.

consumed feedstuffs contaminated with aflatoxin B1.

Recent reports of the International Agency for

Research on Cancer (IARC) assessed the potential

carcinogenicrisks to

Organization (WHO) and IARC 2002) of aflatoxin

M1, and concluded that even very low levels of

exposure to aflatoxins, i.e. 1ngkg?1bwday?1or less

contribute to a risk of liver cancer (Byrne 2000). Due it

high human risks the European Union has established

a very restrict maximum residue limit (MRL) or

maximum permitted limit (MPL) for aflatoxin M1

content in milk at 0.050mg kg?1(Commission of the

European Communities 1998, 2001, 2004, 2010).

European Union legislation (Commission of the

European Communities 2006) states that the uncer-

tainty of the analytical measurement for aflatoxin M1

contamination should be considered when reporting

and interpreting the analytical results (Commission of

the European Communities 2002, 2004, 2006). Besides,

European Union legislation (Commission of the

European Communities 2006) establishes the necessity

of declaring the correction of the aflatoxin M1

contamination by the recovery, or not, and what is

the recovery. It also establishes the minimum perfor-

mance criteria that an analysis procedure should meet

to be used in the analysis of aflatoxin M1in milk that

was used when setting criteria for method perfor-

mance. MERCOSUR and Brazil (Age ˆ ncia Nacional de

Vigila ˆ ncia Sanita ´ ria (ANVISA) 2011; MERCOSUR/

GMC/RES 2002) have established limits of 0.5 and

5mgkg?1for aflatoxin M1in milk and powdered milk

along sampling plans (Mercosur), but no method

performance has been established, nor the necessity

of reporting the recovery or uncertainty of the analyt-

ical measurement. Recovery and measurement uncer-

taintyhavebeenreported

customers or according to institution policies as

defined by ISO/IEC 17025 (ISO 2005).

The LACQSA/LANAGRO-MG/MAPA labora-

tory is accredited according ISO/IEC 17025 by the

national metrology institute – the Instituto Nacional

de Metrologia Normalizac ¸ a ˜ o e Qualidade Industrial

(INMETRO) – to realise the analytical procedure

for the determination of the herein reported ‘determi-

nation of aflatoxin M1in milk by visual and densito-

metricthin-layerchromatography

immunoaffinity column clean-up’.

humans (WorldHealth

when demandedby

(TLC)with

Materials and methods

Uncertainty estimation procedure

In the cause-and-effect approach for uncertainty esti-

mation, or bottom-up approach, the detailed knowl-

edge of the chemical analytical procedure is necessary

to enable the correct identification of the measurand

and of its sources of uncertainties. In this paper the

680K.L. Carvalho et al.

Page 3

four steps presented elsewhere were strictly followed

(Carvalho, Santos, et al. Forthcoming).

Step 1: Measurand specification

Measurand specification is realised through the mea-

surand equation and the measurement procedure

(ABNT, INMETRO 2003; BIPM et al. 1995; Ellison

et al. 2000) or analytical procedure, which is normally

documented through the standard operation proce-

dures. The analytical procedure for aflatoxin M1

determinationusing TLC

Complete measurandequations,

below, show in detail all the direct measurements

realised during the analytical procedure using TLC

with densitometric quantification (Equation 2) or with

visual quantification (Equation 3).

issummarised

also

below.

presented

Analytical procedure

The analytical procedure for aflatoxin M1determina-

tion was adapted from a published immunoaffinity

clean-up method (Dragacci et al. 2001). It can be

summarised in the following steps:

(1) A volume of 100ml (Vs) of defatted milk was

taken after centrifugation and filtered by using

folded filter paper.

(2) Clean-up was done by using an immunoaffinity

column (VICAN or R-Biopharm), washing

with 5ml of water, followed by the elution of

aflatoxin M1 with a solution of 2.5ml of

acetonitrile and methanol (2:1, v/v) and 2.5ml

of pure methanol. The sample eluate was

evaporated at 40?C under nitrogen bubbling

just before complete dryness (critical step).

(3) The dried extract was re-dissolved with the

addition of 100ml (Vr) of toluene–acetonitrile

(9:1 v/v) and by homogenisation with sonica-

tion and vortexed mix.

(4) Application of 20ml (Va) of the sample extract

obtained in step 3 and of 10ml (Vp) of the

calibration aflatoxin M1standard solutions (six

spots) was made on the TLC plate, followed by

elution with ether–methanol–water (96:3:1 v/v/

v).

(5) Detection and quantification step: the fluores-

cence intensities of the spots for the test

samples and the calibration solutions were

read under ultraviolet light (?¼365nm) by

visual comparison between sample spots and

calibration standard solutions spots or by using

a densitometer (dual wavelength flying spot

scanning densitometer;

Shimadzu, Sa ˜ o Paulo, Brasil) at a wavelength

of 365nm.

modelCS-9301PC,

Step 2: Identifying the sources of uncertainties

As proposed elsewhere (Carvalho, Santos, et al.

Forthcoming) the cause-and-effect, or Ishikawa, dia-

gram was constructed by considering the input quan-

tities that appear in the measurand equation as the

primary sources of uncertainties. Other sources of

uncertainties considered are the allowed range of

uncorrected bias due to the allowed variation of the

recovery ratio (RR), and the intermediary precision or

internal reproducibility or intra-laboratory reproduc-

ibility obtained from the analysis of control samples

for quality control.

Volumes

For the measurements of volume, four basic sources of

uncertainties were considered: instrumental uncer-

tainty (Joint Committee for Guides in Metrology

(JCGM/WG 2) et al. 2008; INMETRO 2009) due to

the formal calibration of the instrument of volume

measurement; uncertainty due to resolution for grad-

uated volume instruments; uncertainty due to the

variation of the laboratory temperature around the

mean laboratory temperature; and uncertainty due to

volume measurement repeatability.

Calibration

For TLC calibration, six standard calibration solutions

were used labelled P1,P2,...,P6with concentrations

near to 1.0, 0.5, 0.3, 0.2, 0.1 and 0.05mgml?1,

respectively. As presented by Carvalho, Santos, et al.

(Forthcoming), the resolution of the visual calibration

was modelled by considering the different intervals

(D’s) among the concentrations of the calibration

solutions. This model considers that when reading,

by visual comparison, the analyst can state that the

fluorescence of the sample spot is equal to the

fluorescence of one of the calibration standard solu-

tions, Pi, or that its fluorescence is found between two

successive solutions Pi–Piþ1. This model is represented

in Figure 1. From this model the uncertainty of the

visual quantification due to calibration resolution

• • • • •

• • • • •

• • • • •

P6P5P1P2

P3

P4

P1P2P2P3P3P4

Δ(P1-P2)

Δ(P1P2-P2P3)

Δ(P3-P4)

Δ(P2-P3)

Δ(P2P3-P3P4)

Δ(P4-P5)

Figure 1. Variations (D) between any pair Pi–Piþ1 of

standards and between half readings PiPiþ1–Piþ1Piþ2 used

to model the uncertainty due to the resolution of the visual

calibration of the fluorescence readings in the TLC.

Food Additives and Contaminants681

Page 4

(uVCresol) was estimated for different fluorescence

readings as a function of the standard calibration

solution concentration, C(Pi) by using the rectangular

probability density function (PDF) of the following

equations (1a) and (1b):

?

The uncertainty obtained by Equation (1a) is

attributed to the averaged concentration of the stan-

dard solutions Piand Piþ1:

C Pi

ð

uVCresol

2

???

The uncertainty obtained by Equation (1b) is

attributed to the concentration:

uVCresol

CðPiÞ þ CðPiþ1Þ

2

?

¼

CðPiÞ ? CðPiþ1Þ

2

????

ffiffiffi

3

p

ð1aÞ

Þ þ C Piþ1

?

CðPiÞþCðPiþ1Þ

ð

?

Þ½?=2

CðPiÞ þ CðPiþ1Þ

þCðPiþ1Þ þ CðPiþ2Þ

???

2

??

2

?

¼

2

?CðPiþ1ÞþCðPiþ2Þ

23

2

ffiffiffi

p

ð1bÞ

f½CðPiÞ þ CðPiþ1Þ?=2 þ ½CðPiþ1Þ þ CðPiþ2Þ?=2g=2

For each visual reading, the modelled resolution

uncertainty was combined with the reproducibility

standard deviation due to the two or three readings of

the TLC spots fluorescence intensities of different

analysts.

For the densitometric TLC calibration, the areas of

the calibration peaks, associated with the fluorescence

of each standard calibration solution spots on the TLC

plate, were fitted against the standard calibration

solutionconcentrationusing

square (WLS). To estimate the uncertainties on the

densitometric TLC peak areas, the results of the

calibration curves obtained over 8 months were used.

The areas of the densitometric peaks of each of four

sets of calibration standard solutions were measured

on at least 3 days to obtain a model for the sample

standard deviations (square root of the variances) of

these areas as a function of the standard calibration

solution concentration. This model was used to weight

each densitometric peak area in the WLS fitting of the

calibration curves (see the last column of Table 2).

With this approach, the uncertainties of the intercept

and slope of the calibration curves take into account

the sources of uncertainties due to the intermediary

precision of the densitometric peak areas as well as,

and implicitly, that due to the preparation of the

standardcalibrationsolutions,

EuropeanUnionnorms

European Communities 2002, 2004). Although these

European laws do not specifically apply to mycotoxin

analysis, the use of the above approach to estimate

calibration uncertainty is consistent with statistical

concepts and is recommended in order to avoid the

theweightedleast

asrequiredby

the(Commissionof

underestimation of this source of uncertainty. This

model does not consider the intrinsic Poisson uncer-

tainty of the densitometric peak area of each individual

spot, which is negligible compared with the other two

sources of uncertainties considered.

Recovery

The analytical method presents a recovery ratio

variable from day to day, but no correction is applied

to the results. Therefore, the allowed recovery ratio

range was used to estimate a Type B uncertainty, as

described by Carvalho, Santos, et al. (Forthcoming).

Intermediary precision

Although the uncertainty of repeatability was included

in the volume and calibration uncertainty sources, it

also included an intermediary precision to account for

other sources of uncertainties from bath-to-bath, day-

to-day, sample preparation-to-sample preparation, etc.

variations.

Step 3: Estimating (quantifying) the standard

uncertainties of each source of uncertainty

The uncertainties due to volume measurements,

visual anddensitometric

quantificationwereestimated

Carvalho, Santos, et al. (Forthcoming).

TLC calibration

as described

and

by

Recovery

When recovery is not used to correct the results, its

uncertainty is obtained from the allowed variation of

the recovery of the quality control samples analysed

with each bath of analysis. For the samples with

aflatoxin M1contamination from 0.01 to 0.05mgl?1,

the allowed recovery ratio range is from 60% to 120%;

above this contamination this allowed range becomes

from 70% to 110% (Commission of the European

Communities 2006). Each of these ranges defines a

total amplitude 2a of the rectangular PDF for a Type B

estimation of the uncertainty.

Intermediary precision

The uncertainties of intermediary precision of the

densitometric and visual TLC methods were obtained

from the straight line model fitted to the data of the

standard deviation of the results of analysis of the

fortified blank samples with 0.02, 0.05 and 0.5mgl?1

for quality control, which was carried out over a period

of 11 months. This is a Type A estimation; and as only

three points were used to fit the straight line, the

degrees of freedom of this uncertainty component

was only 1.

682K.L. Carvalho et al.

Page 5

Step 4: Calculating the combined and expanded

uncertainty of the measurand

To calculate the combined standard uncertainty of the

concentration of aflatoxin M1in the test sample, the

classical uncertainties propagation law was used. To

obtain the expanded combined uncertainty, the stan-

dard combined uncertaity is multiplied by the coverage

factor to a coverage probability of 95%, as described

by Carvalho, Santos, et al. (Forthcoming). The cover-

age factor is obtained from the student PDF point of

probabilities, depending on the effective degree of

freedom calculated by the Welch-Satterthwaite equa-

tion (INMETRO 2009; Joint Committee for Guides in

Metrology (JCGM/WG 2) 2008; Carvalho, Santos,

et al. Forthcoming).

Results and discussion

Step 1: Measurand specification

After a detailed analysis of the standard operation

procedures the proposed measurand equations were:

For densitometric TLC quantification:

CaflaM1 ¼Vp ? CSAA ? Vr

¼Vp ? APDA ? a

Va ? Vs

ð

b ? Va ? Vs

? CFrecupþ Cprecint

Þ ? Vr

?

1

RRþ Cprecint

ð2Þ

For visual TLC quantification:

CaflaM1 ¼Vp ? LVm þ CResol

? CFrecupþ Cprecint

where CaflaM1 is the aflatoxin M1 concentration

(mgl?1or ppb) quantified in the milk sample; CSAA

is the concentration (mgml?1or ppm) of the test sample

solution applied in the TLC plate; Vp is the volume (ml)

of the standard solutions of different concentrations

applied to the TLC plate to quantify the aflatoxin

contamination by using a densitometer calibration

curve or by visual comparison of the fluorescence

intensity related to aflatoxin M1from the standards

and the test sample spots; Vr is the volume (ml) of the

solvent used to re-dissolve the sample after drying the

test sample extract obtained after the sample clean-up

through the elution of aflatoxin M1

immunoaffinity column; Va is the volume (ml) of the

test sample solution applied to the TLC plate; Vs is the

test sample volume (ml) of the milk injected onto

the immunoaffinity column to be purified; APDA is the

area for the densitometric peak for the test sample

solution applied in the TLC plate; a and b are,

respectively, the intercept and slope of the densitom-

eter calibration curve; CFrecup¼1/RR is the correction

factor of the unitary value due to the uncorrected

ð Þ ? Vr

Va ? Vs

ð3Þ

from the

recovery; RR is the recovery ratio measured through

the analyses of a spike sample of quality control during

the bath of analysis; CResolis a null correction applied

to the average reading LVm due to the resolution of

the visual calibration (by introducing this null correc-

tion the not null uncertainty due to the visual reading

resolution was computed); Cprecintis the null correction

due to the intermediary precision; and LVm is the

mean visual reading for the test sample concentration

(mgml?1or ppm) obtained by the average of three

individual readings of three different analysts for the

fluorescence intensity of the test sample spot compared

with the fluorescence of the six spots of the aflatoxin

M1standard calibration solutions.

Note that CSAA is obtained from the intercept and

slope of the calibration curve by the equation:

CSAA ¼APDA ? a

b

ð4Þ

Therefore, the uncertainty of CSAA, u(CSAA), will

be dependent on the uncertainties and covariance of a

and b, but also on the repeatability uncertainty of the

instrumental response for the test sample solution

applied to the TLC plate, u(APDA).

Step 2: Identifying the sources of uncertainties

The cause-and-effect (Ishikawa) diagrams, obtained

for densitometric and visual methods, are showed in

Figures 2 and 3, respectively. All the input quantities

in Equations (2) and (3) are present in these

cause-and-effect diagrams. For Figures 2 and 3 some

symbols are defined as above; other symbols are as

follows: CP, concentration of the calibration standard

solutions; APDP, area of the densitometric peak for

the spots of the calibration standard solutions; Calibr,

calibration; Resol, resolution of the measurement

instrument; Repet, repeatability; and Lab Temp Var,

variation of the laboratory temperature around its

mean value.

Step 3: Estimating (quantifying) the standard

uncertainties of each source of uncertainty

Volume

The three Type B uncertainties of the volume

measurements due to the resolution, uResol(V), the

calibration,uCalib(V), and

temperature variation, DT, of 5K around the mean

laboratory temperature, uVartemp(V), as well as the

Type A uncertainty due to the repeatability, uRep(V),

were obtained as shown elsewhere (Carvalho, Santos,

et al. Forthcoming).

The measurement function (INMETRO 2009; Joint

Committee for Guides in Metrology (JCGM/WG 2)

thetotallaboratory

Food Additives and Contaminants 683

Page 6

2008)

quantity is:

ormeasurandequationofthevolume

V ¼ VnominalþCResolþ CCalibþ CVartempþ CRep

By applying the law of uncertainty propagation on

Equation (5) the combined uncertainties of each of the

four measured volumes, V (Vs, Vr, Va or Vp), were

obtained by the equation:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð5Þ

u V

ð Þ ¼

u2

ResolV

ð Þ þ u2

CalibV

ð Þ þ u2

VartempV

ð Þ þ u2

RepV

ð Þ

ð6Þ

q

Table 1 shows the metrological characteristics of

the instruments used for volume measurement: their

standard uncertainties due to instrument resolution,

calibration, repeatability and laboratory temperature

variation, as well as the combined uncertainty of

volumemeasurementu(V)

Equation (6).

ascalculatedby

Calibration

The uncertainty of the concentration of the aflatoxin

M1test sample solution, u(CSAA), applied to the TLC

plate determined by the densitometric quantification

came from the uncertainties u(a)¼saand u(b)¼sb, and

from the covariance u(a,b)¼cov(a,b) of the intercept

and of the slope of the calibration straight line

(Carvalho, Gonc ¸ alves, et al. Forthcoming, Carvalho,

Santos, et al. Forthcoming).

To quantify a test sample, a set of standard

calibration solutions with the concentrations presented

in Table 2 was used. The same set of calibration

standard solutions was used for visual and densito-

metric TLC quantification. The peak areas of these

calibration standard solutions for a typical calibration

are also shown in the Table 2.

The uncertainties (sample standard deviations) of

the APDP in the last column of Table 2 and the

uncertainties bars in Figure 4 clearly show the

heteroskedasticity of the instrumental responses, and

were experimentally obtained as presented above

through the linear model:

uðAPDPÞ ¼ 10:80037 þ 165:56443 ? CP

By fitting a straight line to the data of the Table 2

using the WLS (the continuous line in Figure 4), the

intercept a¼(7.83?17.00), the slope b¼(1767.18?

ð7Þ

VpVr Vs

Volumes

Purification

CSAA

Experimeters

Allowed recovery variation

range

APDA

a

APDP

CP

uc(CaflaM1)

and

U(CaflaM1)

Va

Resol

Lab Temp Var

Calibr

Repeat

Calibration

b

Different days

Intermediary precision or internal

reproducibility or intralaboratorial

precision

Figure 2. Cause-and-effect (Ishikawa) diagram to represent the sources of uncertainties to estimate the uncertainty of the

aflatoxin M1content u(CaflaM1) for densitometric TLC quantification.

u(CaflaM1)

and

U(CaflaM1)

Vp VrVs

Volumes

Intermediary precision or internal

reproducibility or intralaboratorial

reproducibility

Purification

LVm

Calibration

Experimeters

Different days

Allowed recovery variation

range

Va

Resol

Lab Temp Var

Calibr

Repeat

ResolutionRepeatabilty

Figure 3. Cause-and-effect (Ishikawa) diagram to represent the sources of uncertainties to estimate the uncertainty of the

aflatoxin M1content u(CaflaM1) for visual TLC quantification.

684K.L. Carvalho et al.

Page 7

115.91) mlmg?1and the covariance between the inter-

cept and the slope cov(a,b)¼?1248.1mlmg?1were

obtained. Theintercept standard

17mlmg?1shows that it is statistically equal to zero.

For comparison, the intercept and the slope of the

calibration line obtained from the ordinary least square

(OLS) fit (dash and dotted line in Figure 4) were

a¼24.73 and b¼1704.2mlmg?1. The uncertainties of

these OLS fitted parameters were not reported because

they do not have any statistical meaning once the

residual standard deviation of the OLS fit does not

uncertainty of

represent the uncertainty of any instrumental response

within the calibration concentration range. The WLS

and OLS lines cross each other at the crossing point

(0.268mgml?1) marked in the Figure 4 with a signal

plus (þ). At concentrations lower than 0.268mgml?1,

before the crossing point in Figure 4, the WLS line is

below the OLS line; after this concentration the WLS

line is above. This implies that the use of the OLS fit in

the present case generates a proportional negative bias

for milk samples contaminated with aflatoxin M1at

levelsof contamination

2¼0.134mgml?1and a positive bias above this con-

tamination. Figure 4 also shows the upper, UPL

(dashed line), and lower (dotted line), LPL, prediction

limits (lines) for 95% of confidence as obtained by the

WLS. Due the linear heteroskedasticity they are not

symmetrical around the centroid (median point, arith-

metic average), as occurs in the OLS, but are around

the barycentre (weighted centroid, weighted average),

where the prediction limit lines are the nearest to the

WLS fitted straight line. The region around the

barycentre of the calibration line is of better precision.

The area of the densitometric peak for a test sample

solution was 201.082, leading, according to Equation

(4) and the WLS calibration fitting parameters, to

CSAA¼0.1094mgml?1and, according to Equation

(2), to an aflatoxin M1content of 0.05468mgl?1. This

content on the basis of the OLS fit is 5.4% less than the

content obtained from the WLS statistical method.

The barycentre and the centroid, which are marked

in Figure4,arethe

(0.0929mgml?1,172) and

respectively.

Using the curves of the limit of prediction, as

recommended by the standard ISO 11843 (ISO 1997,

2000) and described by Carvalho, Santos, et al.

(Forthcoming), thedecision

0.1171mgml?1(see CC? in Figure 4) was obtained,

corresponding, according to Equation (2), to aflatoxin

M1contamination CC?¼0.059mgl?1. Using the sim-

plifiedmethodology recommended

Commission Decision657

European Communities 2002), a decision limit of

CC?¼0.1132mgml?1corresponding to an aflatoxin

lowerthan 0.268/

points withcoordinates

(0.3574mgml?1,634),

limitCC?¼

in European

of(Commissionthe

Table 1. Metrological characteristics of the instruments used for volume measurements in TLC quantification of aflatoxin M1:

scale division (d), scale space (SS), kdis the divisor in the equation: uResol(V)¼SS/[kdx

calibration laboratory temperature variation and repeatability, as well as the combined standard uncertainty for graduated

cylinder (Vs), automatic pipette (Vr) and microsyringe (Va and Vp).

ffiffiffi

3

p

], uncertainties due to resolution,

Quantity ValuedSSkd

uResol(V)uCalib(V)uVartemp(V)uRep(V)u (V)

Vs

Vr

Va

Vp

100ml

100ml

20ml

10ml

1ml

1ml

0.5ml

0.5ml

1.6mm

–

1.5mm

1.5mm

4

?

4

4

0.14ml

0.29ml

0.072ml

0.072ml

0.51ml

0.26ml

0.085ml

0.080ml

0.030ml

0.030ml

0.0061ml

0.0030ml

0.31ml

0.001ml

0.001ml

0.001ml

0.61ml

0.39ml

0.11ml

0.11ml

Figure 4. Calibration curve of the densitometric TLC

quantification as obtained by WLS (continuous line) and

OLS fitting (dotted-dashed line). UPL and LPL are the upper

and the lower prediction limit curves for 95% of confidence.

Table 2. Calibration

quantification.

dataoftheTLCdensitometric

StandardCP (mgml?1)APDPu (APDP)

P6

P5

P4

P3

P2

P1

0.036802

0.101913

0.184010

0.331218

0.496827

0.993655

71.019

188.972

342.176

584.259

920.293

1696.175

16.893

27.674

41.266

65.638

93.057

175.314

Note: CP, concentration of the calibration standard solution;

APDP, area of the densitometric peak for the spots of the

calibration standard solutions.

Food Additives and Contaminants685

Page 8

M1contamination of CC?¼0.055mgl?1was found.

Note that these decision limits are nearly 10–20%

larger than the MPL of the European Union.

However, these calculated CC? have only taken into

account the calibration curve uncertainty. By including

the within-laboratory reproducibility they will become

higher. According to European legislation concerning

residues in animals and animal products for human

consumption, samples with residue levels above the

limit of decision will be considered as being non-

compliant (Commission of the European Communities

2002, 2004; ISO 1997). Once, even taking into account

the measurement uncertainty, such samples exceed the

maximum residue limit (MRL) beyond reasonable

doubt (Commission of the European Communities

2002, 2004). Nowadays the decision limit is not applied

to decide about the compliance of the products

contaminated with aflatoxins. However, due to its

statistical consistency, it is expected that in the near

future the decision limit will also apply to mycotoxins

as well as to all other products and contaminants.

In the visual quantification the readings of three

analysts for the fluorescence intensity of a sample spot

were equivalent to those spots of the standard P5,

between the standards P5 and P6 (mean concentration

of P5 and P6), and the standard P5 again. These

readings lead to a mean standard concentration

LVm¼0.0911mgml?1

repeatability¼0.0108mgl?1. This LVm corresponds,

according to Equation (3), to an aflatoxin M1con-

tamination of 0.0455mgl?1.

This contamination, determined by the visual TLC

quantification, was 16.7% lower than that of the

densitometric TLC quantification. For other contam-

ination levels the differences between these methods is

nearly between 15% and 20% and the signal of these

differences alternates randomly, and both analytical

procedures can be considered to be equally accurate

concerning its trueness.

The uncertainty of the mean read of the spot

fluorescence of the test sample in the visual TLC

quantification has two components: repeatability and

resolution. The first was obtained as the standard

deviation of three readings. To obtain a mathematical

model for the uncertainty due to the resolution of

the visual TLC calibration according to Figure 1 and

the strategy presented by Carvalho, Santos, et al.

(Forthcoming), Table 3 was constructed.

Figure 5 shows the linear and parabolic (the

numerator in Equation 8) functions fitted to the data

of DCP against CP of Table 3. Owing to the coefficient

of determination, R2, the parabolic function is only

slightly better than the linear one. However, the

parabolic function fits the data better at low concen-

trations,nearthestandard

0.1mgml?1, which correspond to the maximum per-

mitted contamination level of 0.05mgl?1(European

andstandard deviation of

concentration of

Commission 2010). The fourth column of Table 3

shows the calculated standard concentration variation

around each possible reading of the visual calibration

according to the parabolic model, which is used as the

range of the rectangular PDF (Rr) to estimate the

uncertainty of resolution of the mean visual reading,

uResol(LVm):

p ¼0:447CP2þ0:315CPþ0:025

uResolLVm

ð Þ¼Rr

2

ffiffiffi

32

ffiffiffi

3

p

ð8Þ

where CP is the concentration of a real or hypothetical

standard calibration solution at the mean visual

reading.

y = 0.672x – 0.023

R² = 0.951

y = 0.447x2+ 0.315x + 0.025

R² = 0.969

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.00 0.200.40 0.600.80

Δ ΔCP

CP = Standard Concentration

Concentration variation between neighbours

standard solutions

Figure 5. Linear and the parabolic mathematical functions

modelling the aflatoxin M1concentration variation, DCP,

betweensuccessivecalibration

and between successive half calibration solution concentra-

tions (mean between two standards) used to estimate the

uncertainty of resolution of the visual TLC calibration.

solutionconcentrations

Table 3. Modeling the typical uncertainty of resolution of

the visual TLC calibration.

Standard

solution

CP

(mgml?1)

DCPRruResol(LVm)

P1

Mean of P1P2

P2

Mean of P2P3

P3

Mean of P3P4

P4

Mean of P4P5

P5

Mean of P5P6

P6

1.000

0.750

0.500

0.400

0.300

0.250

0.200

0.150

0.100

0.075

0.050

–0.787

0.513

0.294

0.222

0.160

0.132

0.106

0.082

0.061

0.051

0.042

0.227

0.148

0.0849

0.0641

0.0462

0.0381

0.0306

0.024

0.018

0.015

0.012

0.500

0.350

0.200

0.150

0.100

0.100

0.100

0.075

0.050

–

Note: CP, typical standard concentration; DCP, concentra-

tion variation between neighbouring calibration solutions;

Rr¼2a is the range of the rectangular PDF. In the second

column, values shown in bold are the concentrations of real

standard calibration solutions; other values correspond to

half readings.

686K.L. Carvalho et al.

Page 9

Recovery

As mentioned above, the uncertainty due to the

recoveryvariation assumes

depending on the contamination of the test sample.

The correction factor due to the recovery (CFrecup) is

equal to the inverse of the recovery ratio. For samples

with contamination in the range 0.01–0.05mgl?1,

for which the allowed range of the recovery ratio

was 60–120%, the standard uncertainty of the recov-

ery ratio u(RR) was estimated by (Carvalho, Santos,

et al. Forthcoming, Carvalho, Gonc ¸ alves et al.

Forthcoming):

twodifferentvalues

u RR

ð Þ ¼1:20 ? 0:60

2

ffiffiffi

3

p

¼ 0:1732

ð9Þ

As CFrecup¼1/RR, the uncertainty of the correc-

tion factor of recovery u(CFrecup) is (Carvalho,

Gonc ¸ alves et al. Forthcoming, Carvalho, Santos,

et al. Forthcoming):

u CFrecup

??¼u RR

ðÞ

RR2¼1:20 ? 0:60

2

ffiffiffi

3

p

? RR2

ð10Þ

While for samples with contamination above

0.05mgl?1, for which the allowed range of recovery

was 70–110%, the standard uncertainty of the recovery

ratio u(RR) was estimated by (Carvalho, Gonc ¸ alves

etal.Forthcoming,Carvalho,

Forthcoming)

Santos,etal.

u RR

ðÞ ¼1:10 ? 0:70

2

ffiffiffi

3

p

¼ 0:1155

ð11Þ

u CFrecup

??¼u RR

ðÞ

RR2¼1:10 ? 0:70

2

ffiffiffi

3

p

? RR2

ð12Þ

As the recovery ratio varies significantly even

within a batch of analysis, the median value 0.90 of

the allowed range of recovery ratio variation was used

for both the recovery ratio range in Equations (10)

and (12), leading to values of 0.740741/(2

0.21383mgl?1and of 0.493827/(2

as standard uncertainties for the correction factor of

the recovery ratio.

ffiffiffi

3

p

)¼

ffiffiffi

3

p

)¼0.14256mgl?1

Intermediary precision

The straight line fitted to three standard deviations of

the spiked samples blanks with aflatoxin M1standard

solutions for 0.02, 0.05 and 0.5mgl?1aflatoxin M1

contaminations, measured many times during the 11

months, leads to the following models for uncertainty

due to intermediary precision or intra-laboratory

reproducibility:

For the densitometric TLC quantification:

u precint

ð Þ ¼ 0:2262 ? CaflaM1 mg=kg

ðÞð13Þ

For the visual TLC quantification:

u precint

ðÞ ¼ 0:0006 þ 0:2706 ? CaflaM1 mg=kg

ð Þ ð14Þ

Step 4: Calculating the combined and expanded

uncertainty of the measurand

To combine the standard uncertainties of the input and

influence quantities, u(?), the particular case of the

uncertainty propagation law by propagating the rela-

tive uncertainties was not used, as often occurs (Ellison

et al. 2000; Barwick and Ellison 2000a, 2000b;

Armishaw 2003; Calaresu et al. 2006), because the

present measurand equations do not only have multi-

plications and division operations. For densitometric

TLC quantification the uncertainty propagation law

takes the form:

u caflaM1

ð

v

u

u

u

t

v

u

u

u

u

u

u

Þ

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

h

þ

?

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

h

þ

h

þ

@CaflaM1

@Vp

u Vp

ðÞ

h

þ

i2þ

Þ

i2

@CaflaM1

@CSAAu CSAA

ðÞ

hi2

@CaflaM1

@Vr

u Vr

ð

i2þ

@CaflaM1

@Va

u Va

ðÞ

hi2

@CaflaM1

@Vs

u Vs

ðÞ

h

h

þ

@CaflaM1

@CFrecupu CFrecup

?

i2þ

@CaflaM1

@Cprecintu Cprecint

??

hi2

u

u

u

u

u

u

u

u

u

u

u

u

u

u

u

u

u

u

t

¼

@CaflaM1

@Vp

u Vp

ðÞ

h

þ

i2þ

i2þ

Þ

i2

@CaflaM1

@APDAu APDA

ðÞ

h

h

i2

@CaflaM1

@a

u a ð Þ

@CaflaM1

@b

u b ð Þ

i2

@CaflaM1

@Vr

u Vr

ð

hi2þ

@CaflaM1

@Va

u Va

ðÞ

hi2

@CaflaM1

@Vs

u Vs

ðÞ

@CaflaM1

@CFrecupu CFrecup

??

ð

hi2þ

@CaflaM1

@Cprecintu Cprecint

??

hi2

þ2@CaflaM1

@a

@CaflaM1

@b

u a,b

Þ

u

u

u

u

u

u

u

u

u

u

u

u

ð15Þ

The first and second forms of the above equation

are based on the respective forms of the measurand

equation presented in Equation (2). Here the partial

derivatives (differential quotient or differential coeffi-

cient) are the sensitivity coefficients given by the

following equations:

@CaflaM1

@Vp

¼

APDA ? a

b ? Va ? Vs

ðÞ ? Vr

? CFrecup¼CaflaM1

Vp

ð16Þ

Food Additives and Contaminants 687

Page 10

@CaflaM1

@Va

¼ ?Vp ? APDA ? a

b ? Va2? Vs

¼ ?CaflaM1

Va

ðÞ ? Vr

? CFrecup

ð17Þ

@CaflaM1

@Vr

¼Vp ? APDA ? a

b ? Va ? Vs

¼CaflaM1

Vr

ðÞ

? CFrecup

ð18Þ

@CaflaM1

@Vs

¼ ?Vp ? APDA ? a

b ? Va ? Vs2

¼ ?CaflaM1

Vs

ðÞ ? Vr

? CFrecup

ð19Þ

@CaflaM1

CFrecup

¼Vp ? APDA ? a

b ? Va ? Vs

ðÞ ? Vr

¼CaflaM1

CFrecup

ð20Þ

@CaflaM1

@Cprecint

¼ 1

ð21Þ

@CaflaM1

DAPDA

¼

Vp ? Vr

b ? Va ? Vs? CFrecup¼

CaflaM1

APDA ? a

ð22Þ

@CaflaM1

@a

¼ ?

Vp ? Vr

b ? Va ? Vs? CFrecup¼ ?CaflaM1

APDA ? a

ð23Þ

@CaflaM1

@b

¼ ?Vp ? APDA ? a

b2? Va ? Vs

¼ ?CaflaM1

b

ðÞ ? Vr

? CFrecup

ð24Þ

The only covariance taken into account was that

between the intercept and the slope, u(a,b)¼cov(a,b),

of the calibration curve of the densitometric TLC

quantification obtained from WLS fitting.

For visual TLC quantification, the uncertainty

propagation law takes the form:

u caflaM1

ð

v

u

u

u

t

Þ

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

h

þ

?

@CaflaM1

@Vp

u Vp

ðÞ

h

þ

i2þ

Þ

i2þ

@CaflaM1

@LVmu LVm

ðÞ

i2

hi2

@CaflaM1

@Vr

u Vr

ð

i2þ

@CaflaM1

@Va

u Va

ðÞ

h

h

?

@CaflaM1

@Vs

u Vs

ðÞ

h

h

@CaflaM1

@CResolu CResol

i2þ

ðÞ

i2

þ

@CaflaM1

@CFrecupu CFrecup

@CaflaM1

@Cprecintu Cprecint

??

hi2

ð25Þ

u

u

u

u

u

u

u

u

u

u

u

u

where the partial derivatives (differential quotient or

differential coefficient) are the sensitivity coefficients

given by:

@CaflaM1

@Vp

¼

¼CaflaM1

Vp

LVm ? CResol

Va ? Vs

ðÞ ? Vr

? CFrecup

ð26Þ

@CaflaM1

@Va

¼ ?Vp ? LVm ? CResol

¼ ?CaflaM1

Va

ðÞ ? Vr

Va2? Vs

? CFrecup

ð27Þ

@CaflaM1

@Vr

¼Vp ? LVm ? CResol

Va ? Vs

¼CaflaM1

Va

ðÞ

? CFrecup

ð28Þ

@CaflaM1

@Vs

¼ ?Vp ? LVm ? CResol

¼ ?CaflaM1

Vs

ðÞ ? Vr

Va ? Vs2

? CFrecup

ð29Þ

@CaflaM1

@CFrecup

¼Vp ? LVm ? CResol

ðÞ ? Vr

Va ? Vs

¼CaflaM1

CFrecup

ð30Þ

@CaflaM1

@Cprecint

¼ 1

ð31Þ

@CaflaM1

@LVm

¼Vp ? Vr

Va ? Vs? CFrecup¼

CaflaM1

LVm ? CResol

ð32Þ

@CaflaM1

@CResol

¼ ?Vp ? Vr

Va ? Vs? CFrecup¼

CaflaM1

LVm ? CResol

ð33Þ

Tables4 and5summarisetheuncertainty

combination for a blank sample spiked with aflatoxin

M1to a content of 0.05mgl?1, which is the European

Union maximum permitted value, and determined as

0.04553mgl?1by visual TLC quantification and as

0.05468mgl?1by densitometric TLC quantification.

The standard uncertainties of the quantities in

Tables 4 and 5, where an asterisk (*) is shown after the

quantity symbol, are obtained through the uncertainty

combination of the uncertainties at the lines immedi-

ately before it. For the case of the volumes, this

combination is realised according to Equation (6). The

graduated cylinder used to measure Vs was the unique

volume measurement instrument not formally cali-

brated. However, the data for internal quality control

of a set of such cylinders showed a maximum bias of

0.8834ml. In a conservative way, this bias was used as

the half range of the rectangular PDF (see the

688K.L. Carvalho et al.

Page 11

Table 4. Calculation of the combined uncertainty for densitometric TLC quantification of aflatoxin M1in milk.

Input or influence quantities

sources of uncertainties

Probability density

function (PDF) (distribution)

Uncertainty contribution

ui(y) or u(y;xi)

Degrees of

freedom ?ior ?eff

[ui(y)]4/?i

Symbol

Value, xi

Interval

Units

Type

Name

Divider, k

Standard

uncertainty u(xi)

Sensitivity

coefficient, ci

ui(y) or u(y;xi)

CResol

0

0.125

ml

B

Rectangular

1.73205

0.072169

0.005468

0.000395

1.0Eþ99

2.4E–113

CVartemp

0

0.00525

ml

B

Rectangular

1.73205

0.003031

0.005468

0.000017

1.0Eþ99

7.5E–119

CCalib

0

0.16

ml

B

Normal

2

0.08

0.005468

0.000437

1.0Eþ99

3.6E–113

CRep

0

0.0003

ml

A

Normal

1

0.0003

0.005468

0.000002

9

8.04E–25

Vp*

10

ml

0.107785

0.005468

0.000589

CResol

0

0.5

ml

B

Rectangular

1.73205

0.288675

0.000547

0.000158

1.0Eþ99

6.2E–115

CVartemp

0

0.0525

ml

B

Rectangular

1.73205

0.030311

0.000547

1.66E–05

1.0Eþ99

7.5E–119

CCalib

0

0.51

ml

B

Normal

2

0.255

0.000547

0.000139

1.0Eþ99

3.7E–115

CRep

0

0.00055

ml

A

Normal

1

0.00055

0.000547

3.01E–07

9

9.08E–28

Vr*

100

ml

0.386364

0.000547

0.000211

CResol

0

0.125

ml

B

Rectangular

1.73205

0.072169

0.002734

0.000197

1.0Eþ99

1.5E–114

CVartemp

0

0.0105

ml

B

Rectangular

1.73205

0.006062

0.002734

1.66E–05

1.0Eþ99

7.5E–119

CCalib

0

0.17

ml

B

Normal

2

0.085

0.002734

0.000232

1.0Eþ99

2.9E–114

CRep

0

0.0006

ml

A

Normal

1

0.0006

0.002734

1.64E–06

9

8.04E–25

Va*

20

ml

0.111671

0.002734

0.0003053

CResol

0

0.25

ml

B

Rectangular

1.73205

0.144338

0.000547

7.89E–05

1.0Eþ99

3.8E–116

CVartemp

0

0.0525

ml

B

Rectangular

1.73205

0.030311

0.000547

1.66E–05

1.0Eþ99

7.5E–119

CCalib

0

0.8834

ml

B

Rectangular

1.73205

0.510031

0.000547

0.000279

1.0Eþ99

6.0E–114

CRep

0

0.3056

ml

A

Normal

1

0.3056

0.000547

0.000167

4

1.94E–16

Vs*

100

ml

0.612597

0.000547

0.000335

APDA

201.082

28.9060

1

A

Normal

1

28.90599

0.000283

0.008179

16

2.79E–10

a

7.83

17.0011

1

A

Normal

1

17.001138

0.000283

0.00481

4

1.33E–10

b

1767.18

115.911

mlmg?1

A

Normal

1

115.911

3.09E–05

0.003586

4

4.13E–11

Cov(a,b)

–1248.1

–1248.1

mlmg?1

A

Normal

1

–1248.1

8.75E–09

–1.09E–05

–

–

CSAA*

0.1094

mgml?1

0.018004

0.5

0.009002

CSAA*

0.1094

mgml?1

0.020287**

0.5

0.010143

CFrecup

1

0.493827

1

B

Rectangular

3.46410

0.142556

0.045529

0.006491

1.0Eþ99

1.7E–108

Cprecint

0

0.01237

mgl?1

A

Normal

1

0.012370

1

0.01237

1

2.34E–08

Sum

2.387E–08

Summary

Quantity

nominal

value Y

Estimated

value y

Combined

standard

uncertainty

uc(y)

Effective

degrees of

freedom ?eff

Coverage

probability

P (%)

Coverage

factor k

Expanded

uncertainty

U(y)

Relative

standard

uncertainty

RSU

Relative

expanded

uncertainty

REU

0.05

0.0547

0.016637

3.21

95

3.1824

0.052945

30.4

96.8

Notes: Only the correlation between the intercept and the slope of the calibration line was considered according to Equation (34).

**Value for the standard uncertainty of CSAA obtained without the covariance between the intercept and slope of the calibration straight line.

Food Additives and Contaminants 689

Page 12

Table 5. Calculation of the combined uncertainty for visual TLC quantification of aflatoxin M1in milk.

Input or influence quantities sources

of uncertainties

Probability density function

(PDF) (distribution)

Uncertainty contribution

ui(y) or u(y;xi)

Degrees of

freedom ?ior ?eff

[ui(y)]4/?i

Symbol

Value, xi

Interval

Units

Type

Name

Divider, k

Standard

uncertainty U(xi)

Sensitivity

coefficient, ci

ui(y) or u(y;xi)

CResol

0

0.125

ml

B

Rectangular

1.73205

0.072169

0.004553

0.000329

1.0Eþ99

1.2E–113

CVartemp

0

0.00525

ml

B

Rectangular

1.73205

0.003031

0.004553

1.380E–05

1.0Eþ99

3.6E–119

CCalib

0

0.16

ml

B

Normal

2

0.08

0.004553

0.000364

1.0Eþ99

1.8E–113

CRep

0

0.0003

ml

A

Normal

1

0.0003

0.004553

1.366E–06

9

3.87E–25

Vp*

10

ml

0.107785

0.004553

0.000491

CResol

0

0.5

ml

B

Rectangular

1.73205

0.288675

0.000455

0.000131

1.0Eþ99

3.0E–115

CVartemp

0

0.0525

ml

B

Rectangular

1.73205

0.030311

0.000455

1.380E–05

1.0Eþ99

3.6E–119

CCalib

0

0.51

ml

B

Normal

2

0.255

0.000455

0.000116

1.0Eþ99

1.8E–115

CRep

0

0.00055

ml

A

Normal

1

0.00055

0.000455

2.504E–07

9

4.37E–28

Vr*

100

ml

0.386364

0.000455

0.000176

CResol

0

0.125

ml

B

Rectangular

1.73205

0.072169

0.002276

0.000164

1.0Eþ99

7.3E–115

CVartemp

0

0.0105

ml

B

Rectangular

1.73205

0.006062

0.002276

1.380E–05

1.0Eþ99

3.6E–119

CCalib

0

0.17

ml

B

Normal

2

0.085

0.002276

0.000194

1.0Eþ99

1.4E–114

CRep

0

0.0006

ml

A

Normal

1

0.0006

0.002276

1.366E–06

9

3.87E–25

Va*

20

ml

0.111671

0.002276

0.000254

CResol

0

0.25

ml

B

Rectangular

1.73205

0.144338

0.000455

6.571E–05

1.0Eþ99

1.9E–116

CVartemp

0

0.0525

ml

B

Rectangular

1.73205

0.030311

0.000455

1.380E–05

1.0Eþ99

3.6E–119

CCalib

0

0.8834

ml

B

Rectangular

1.73205

0.510031

0.000455

0.000232

1.0Eþ99

2.9E–114

CRep

0

0.3056

ml

A

Normal

1

0.3056

0.000455

0.000139

4

9.37E–17

Vs*

100

ml

0.612597

0.000455

0.000279

LVmRepet

0

0.010850779

mg ml?1

A

Normal

1

0.010851

0.5

0.005425

2

4.33E–10

LVmResol

0

0.035885331

mgm l?1

B

Rectangular

1.732051

0.020718

0.5

0.010359

1.0Eþ99

1.2E–107

LV

mg ml?1

0.023388

0.5

0.011694

CFrecup

1

0.740740741

1

B

Rectangular

3.46410

0.213833

0.04553

0.009736

1.0Eþ99

9.0E–108

Cprecint

0

0.012920313

mg l?1

A

Normal

1

0.012920

1

0.012920

1

2.79E–08

Sum

2.83E–08

Summary

Quantity

nominal

value Y

Estimated

value y

Combined

standard

uncertainty

uc(y)

Effective

degrees of

freedom

?eff

Coverage

probability

P (%)

Coverage

factor

k

Expanded

uncertainty

U(y)

Relative

standard

uncertainty

RSU

Relative

expanded

uncertainty

REU

0.05

0.045

0.01997

5.62

95

2.5706

0.051340

36.5263

93.8938

Note: No possible correlations were considered.

690K.L. Carvalho et al.

Page 13

calibration component of Vs in Tables 4 and 5) to

estimate its calibration uncertainty.

The standard uncertainty of the concentration of

the sample solution applied in the TLC plate,

u(CSAA)¼0.018004mgml?1, was obtained from the

WLS parameters of the calibration straight line

through Equation (34) (see also Equation E.3.3 in

Ellison et al. 2000 and Equation 26 in Carvalho,

Santos, et al. Forthcoming):

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

þ2 ? CSAA ? covða,bÞ

u CSAA

ð Þ ¼

u2ðAPDAÞ þ u2ðaÞ þ CSAA2? u2ðbÞ

??

b2

v

t

u

u

u

ð34Þ

As the co-variance (and correlation) between the

intercept and the slope of the calibration straight line is

always negative, u(CSAA)¼0.020287mgml?1(see the

value with a double asterisk in the second line with the

label CSAA in Table 4) without this correlation is

higher than u(CSAA)¼0.018004mgml?1(see the value

in the first line with the label CSAA in Table 4) when

this correlation is taken into account.

The combined standard uncertainties of aflatoxin

M1content as determined by the densitometric and

visual TLC quantification were 0.017 and 0.020mgl?1,

respectively, for a blank sample of milk spiked with

0.05mgl?1(Tables 4 and 5). These combined uncer-

tainties without the contribution of the intermediary

precision are reduced to 0.011 and 0.015mgl?1(see the

bars labelled CU-prec in Figure 6), respectively, almost

equal to the uncertainties of the intermediary preci-

sions alone: 0.0124 and 0.0129mgl?1, respectively.

Figure 6 summarises the calculations in Tables 4

and 5, showing the combined uncertainties and their

contributions to the densitometric and visual TLC

quantifications of a 0.05mgl?1aflatoxin M1 spiked

blank sample milk. For comparison, the values of the

combined uncertainties without the intermediary pre-

cision contribution (see CU-Prec in Figure 6) are

almost equal to the uncertainties of the intermediary

precisions alone. This result corroborates the proposed

use of the intermediary precision as a gross estimation

of the analytical method’s combined uncertainty

(Populaire and Gime ´ nez 2006). The higher contribu-

tions of the combined uncertainty come from the

intermediary precision or within-laboratory reproduc-

ibility, followed by the uncertainties due to the

calibration process and by the uncertainty due to the

allowed recovery variation range. The two last have

nearly the same contribution. The uncertainties due to

the volume measurements are almost negligible and

even the combination of their four contributions is

very low, 0.00144 and 0.00120 for densitometric and

visual quantification, respectively (see the last pair of

bars in Figure 6), corresponding to less than 10% of

the analytical methods’ combined uncertainties. Within

these uncertainties’ sources the most important is the

contribution due the volume measurement of 10ml of

the calibration standard solution, Vp, which is respon-

sible for 41% of the total volume uncertainties in both

methods.

Conclusions

This paper showed in a detailed and conceptually

consistent manner the estimation of the combined

uncertainty of the aflatoxin M1 content in milk,

determined by visual and densitometric quantification

with immunoaffinity column clean-up, through the

cause-and-effect methodology or bottom-up approach

recommended by the ISO through the ISO GUM

(ABNT, INMETRO 2003; BIPM et al. 1995). From

these estimations it seems that the uncertainty of the

densitometric TLC quantification is approximately

between 10% and 30% lower than that of the visual

TLC quantification, but their trueness is the same

(INMETRO 2009; Joint Committee for Guides in

Metrology (JCGM/WG 2) 2008).

The main source of uncertainty in both the visual

and the densitometric methods, as generally happens,

was the intermediary precision, which alone is a

reasonable estimation of the total (combined) analyt-

ical uncertainty (Populaire and Gime ´ nez 2006), which

justifies the use of the top-down approach to estimate

the analytical uncertainty. However, the systematic but

strenuous cause-and-effect (bottom-up) approach for

the estimationof analytical

enabled the authors to understand the most important

sources of uncertainties, which, if reduced, lead to an

efficient analytical method precision improvement.

Excepting the intermediary precision, the uncertainties

due to the calibration process and the allowed recovery

ratio range are equally the most important sources of

uncertainties of the TLC aflatoxin M1analysis, while

the uncertainty contributions due to the volume

methoduncertainty

0.005

0.01

0.015

0.02

Uncertainty contributions

0

Comb Unc CU-precInt Prec calibRecovVolumes

Densitometric

Visual

Figure 6. Contributions for the combined uncertainty of the

visual and densitometric TLC quantification of aflatoxin M1

in milk for a sample blank sample spiked to a content of

0.05mgl?1of aflatoxin M1. For volume measurement, the

bars show the combined contribution of the four measured

volumes.

Food Additives and Contaminants691

Page 14

measurements are practically negligible. The inclusion

of the analytical method uncertainty in the calculation

of the decision limit for product compliance, according

European legislations, increases the importance of a

realistic and conceptually consistent estimation of the

combined and expanded uncertainties of the analytical

procedures. By increasing the analytical precision,

reducing its uncertainty, the consumer and vendor

risks concerning the compliance decision are simulta-

neously decreased.

This work shows that the use of the OLS to fit the

data of measurement instrument calibration, when

they are the subject of heteroskedasticity, leads to

proportional bias and to incorrect estimation of the

measurement uncertainty. The OLS lower and upper

prediction limit lines for the heteroskedastic calibra-

tion data presented in this work are far from the fitted

calibration straight line at the barycentre and MPL

when compared with the WLS lower and upper

prediction limit lines. This implies that the uncertain-

ties of the analyte content at these levels, obtained

from the OLS fit, is much higher than the uncertainty

obtained from the WLS fit. As a consequence, the

decision limit (CC?) of the OLS is also higher than the

CC? of the WLS. This can be a great risk for consumer

health when using the decision limit as a compliance

criterion.

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