Uncertainty Example for Thc 1

Measurement Uncertainty for Drugs – worked example for 9-THC

The calculations relate to this laboratory procedures defined in the associated standard Operation Procedure

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Document ID: Uncertainty example for THC.DOC Date Effective: 16/11/2005 Page 2 of 14

1. Contents

1. Contents_____________________________________________________________ 2

2. Summary ____________________________________________________________ 3

3. Principle_____________________________________________________________ 3

4. Method Procedure_____________________________________________________ 3

5. Standards____________________________________________________________ 3

6. Identified sources of uncertainty_________________________________________ 4

7. Uncertainty of the analysis of �

9-THC in blood by GC/MS.____________________ 5

Uncertainty of stock solution of 9-THC __________________________________________ 5

Uncertainty of dilution - adding 900 L of methanol to 100 L of Stock 9-THC __________ 5

Uncertainty of purity of 9-THC _________________________________________________ 5

Combined Uncertainty of Working Solution of THC U WorkingStd _________________________ 6

8. The Standard Uncertainty of the calibration curve __________________________ 7

Estimation of uncertainty from calibration curves f or this assay U Pred __________________ 7

Estimation of precision uncertainty from Quality Co ntrol Material U QC _________________ 11

Uncertainty associated with mass of test sample U m _______________________________ 11

Combining Uncertainties____________________________ __________________________ 12

Part II. Top Down Approach _______________________________________________ 13

Performance of Quality Controls ____________________ ___________________________ 13

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2. Summary This report describes an uncertainty estimate for the measurement of 9-THC in blood by GC/MS.

3. Principle Specimens of blood are analysed for the determination of 9-THC by GC/MS.

4. Method Procedure 1. Add 1 mL blood (or control or standard) to silanised glass extraction tubes. 2. Add 25 ng d3-THC (25 µL of IS2) to each tube, mix. 3. Add 1.0 mL 1.0 M ammonium sulphate, vortex. 4. Add 7 mL hexane whilst vortexing and extract for 60 min on a wheel. Ensure blood is

mixing with solvent, if not agitate tubes. 5. Centrifuge at 3250 rpm, 10 min. 6. Place tubes in an alcohol freezing bath. When aqueous layer is frozen decant solvent to a

fresh glass extraction tube. 7. Include a ``mix' by adding 100 ng of THC (100 µL WS3) and 25 ng IS (25 µL IS2) to a

separate tube. 8. Evaporate to dryness in a Savant Concentrator (or in heating block under nitrogen). 9. When dry, add 25 µL pentafluoro-1-propanol, vortex, then add 50 µL pentafluoropropionic

anhydride. Stopper and mix on a vortex mixer. 10. Incubate in reaction block at 65 C for 25 min, vortex briefly at 10 min. Evaporate to

dryness under nitrogen. 11. Reconstitute with 100 µL anhydrous ethyl acetate and transfer to MS injection vials

(screw-on with teflon lined septa). 12. Inject 1 µL into the GC/MS. See- Determination of D9-THC in Whole Blood using Gas Chromatography-Mass Spectrometry Hok Chi Chu M and Drummer O.H. Journal of Analytical Toxicology, Volume 26, Number 8, November/December 2002, pp. 575-581(7).

5. Standards Stock solutions of 9-THC are prepared as follows: 1. Prepare dilutions by adding 100 µL STOCK (1 mg/mL) to 900 µL methanol in silanised

glass vials (WS1, 1/10; 100 µg/mL); then � add 100 µL WS1 to 900 µL methanol in silanised glass vials (WS2, 1/100, 10 µg/mL), � add 100 µL WS2 to 900 µL methanol in silanised glass vials (WS3, 1/1000, 1 µg/mL), and � add 100 µL WS3 to 900 µL methanol in silanised glass vials (WS4, 1/10,000, 100 ng/mL), Working Internal Standard 9-THC-D3 2. Prepare dilutions by adding 100 µL STOCK (100 µg/mL) to 900 µL methanol in silanised

glass vials (IS1, 1/10; 10 µg/mL), and add 100 µL IS1 to 900 µL methanol in silanised glass vials (IS2, 1/100, 1 µg/mL).

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6. Identified sources of uncertainty

Result

GC Determination

GC Calibration

Calibration Standards

Extraction

Sample Homogeneity

Sample Mass

Precision

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7. Uncertainty of the analysis of 9-THC in blood by GC/MS.

Uncertainty of stock solution of 9-THC The uncertainty of the stock solution of 9-THC can be determined from the positive displacement pipette which is used to dispense the initial volume of 100 L of stock. � 100 L ± 0.21 % (from manufacturers specifications) � = ± 0.21 L or ±0.00021 mL assuming rectangular distribution USpecification 100� L = 0.00021 3 = 0.0001212 (V1)

Uncertainty of dilution - adding 900 L of methanol to 100 L of Stock 9-THC The uncertainty of the dilution of the stock solution of 9-THC can be determined from the positive displacement pipette which is used to dispense a working volume of 900 L. � 1000 L ± 0.32 % (from manufacturer specifications) � = ±3.2 L or ±0.0032 mL � assuming rectangular distribution USpecification 1000� L = 0.0032 3 = 0.001847 (V2) Also the uncertainty of the pipetter can be calculated from laboratory calibration data which over the last 6 months has shown the relative standard deviation at 1000 L to be 0.50%.

So 22

2121 VVVV UUU +=+ 22 001847.00001212.0

21+=+VVU

000003411.0000000014.021

+=+VVU 00185.0

21=+VVU

Uncertainty of purity of 9-THC Purity of 9-THC is 98.5% min

100 ± 1.5 % 1.0 ± 0.015

assuming rectangular distribution Upurity� 9-THC = 0.015

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3 Upurity� 9-THC = 0.00866 (S)

Combined Uncertainty of Working Solution of THC UWorkingStd Concentration = W / V Where combining all the uncertainties at concentration 100 ng/mL:

( )

( )

2

21

2

1

1

2

100 21100100 ��

�

��

�

++��

�

��

�+�

�

��

�= +

VV

U

V

U

S

UU VVvs where 22

2121 VVVV UUU +=+

( )222

100 0.1

00185.0

1.0

0001212.0

1.0

00866.0��

��

�+�

�

��

�+��

��

�=U

( ) ( ) ( )222100 008849.00866.0001212.0 ++=U

000078304.000749956.0000001468.0100 ++=U

007579.0100 =U

08705.0100 =U

Therefore Combined Uncertainty of Working Solution of THC is 100 g/mL ±±±± 0.09

g/mL

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8. The Standard Uncertainty of the calibration curv e

Estimation of uncertainty from calibration curves for this assay UPred

Eleven different assays run on two different GC/MS systems gave the following results

Table 1. Calculation of response ratios (Y) from known concentrations of 9-THC from 11 calibration curves.

Concentration of 9THC (ng/mL)

1 ng/mL 2 ng/mL 5 ng/mL 10 ng/mL 25 ng/mL 50 ng/mL 100 ng/mL

Replicate Number Response ratio Y

1Y 0.062560621

0.128410915

0.228555167

0.464852795

1.026603852

2.009213006

4.641011772

2 0.049182712

0.092482309

0.201950036

0.420435057

0.999186141

1.995677901

3.76960603

3 0.373704528

0.423910183

0.463017527

0.606303245

1.190583652

2.19849179

4.238783703

4 0.203368982

0.22160709

0.337645914

0.528717476

1.258000153

2.145557351

4.097202842

5 0.048377686

0.093456743

0.22994163

0.41863589

1.130049647

2.248224005

4.440535942

6 0.050155592

0.093162854

0.232216041

0.426423363

1.133886119

2.232503864

4.280899363

7 0.111210007

0.277918549

0.28867392

0.516486273

1.201099446

2.32032062

4.35578943

8 0.045470522

0.092571855

0.221270298

0.440230504

1.100896861

2.229787387

4.500810779

9 0.046593264

0.081692509

0.229806972

0.455622129

0.97563517

2.162878176

4.390148996

10 0.043136229

0.089714551

0.209960181

0.434483445

1.068005632

2.147182608

4.386608706

11 0.050474712

0.088308587

0.217562753

0.420844784

1.081991313

2.131746564

4.689465552

Sum 1.084234855

1.683236144

2.860600441

5.133034961

12.16593799

23.82158327

47.79086312

Mean 0.098566805

0.153021468

0.260054586

0.466639542

1.105994362

2.165598479

4.34462392

n 11 11 11 11 11 11 11 SD 0.103125

071 0.110464

086 0.077985

709 0.059878

251 0.087984

28 0.097810

019 0.255233

348

Response Ratio X

0.04 0.08 0.2 0.4 1 2 4

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Document ID: Uncertainty example for THC.DOC Date Effective: 16/11/2005

Page 8 of 14

Linear Regression calculations for uncertainty asso ciated with calibration curves.

Table 2. Calibration data compiled from 11 calibra tion curves constructed for the determination of 9-THC (using 2, 25, 100 ng/mL points).

Amount X Amount Y X-X (X-X)² Y-Y (Y-Y)² (x-x)(y-y) X² Predicted Y Residuals Standard Residuals

0.04 0.1284109 -1.739469 3.0257524 -1.739469 3.0257527 3.0257526 0.0016 0.1237088 0.0047021 0.0283907

0.04 0.0924823 -1.775398 3.1520367 -1.775398 3.152037 3.1520368 0.0016 0.1237088 -0.031227 -0.188543

0.04 0.4239102 -1.44397 2.0850486 -1.44397 2.0850488 2.0850487 0.0016 0.1237088 0.3002013 1.8125867

0.04 0.2216071 -1.646273 2.7102142 -1.646273 2.7102145 2.7102144 0.0016 0.1237088 0.0978982 0.5911002

0.04 0.0934567 -1.774423 3.1485776 -1.774423 3.1485779 3.1485777 0.0016 0.1237088 -0.030252 -0.182659

0.04 0.0931629 -1.774717 3.1496207 -1.774717 3.1496209 3.1496208 0.0016 0.1237088 -0.030546 -0.184434

0.04 0.2779185 -1.589961 2.5279772 -1.589961 2.5279774 2.5279773 0.0016 0.1237088 0.1542097 0.9311033

0.04 0.0925719 -1.775308 3.1517187 -1.775308 3.151719 3.1517189 0.0016 0.1237088 -0.031137 -0.188002

0.04 0.0816925 -1.786187 3.1904655 -1.786187 3.1904658 3.1904656 0.0016 0.1237088 -0.042016 -0.253691

0.04 0.0897146 -1.778165 3.1618721 -1.778165 3.1618724 3.1618722 0.0016 0.1237088 -0.033994 -0.205254

0.04 0.0883086 -1.779571 3.1668741 -1.779571 3.1668744 3.1668743 0.0016 0.1237088 -0.0354 -0.213743

1 1.0266039 -0.841276 0.7077454 -0.841276 0.7077456 0.7077455 1 1.144687 -0.118083 -0.712975

1 0.9991861 -0.868694 0.7546289 -0.868694 0.754629 0.7546289 1 1.144687 -0.145501 -0.87852

1 1.1905837 -0.677296 0.4587302 -0.677296 0.4587303 0.4587303 1 1.144687 0.0458966 0.2771194

1 1.2580002 -0.60988 0.3719533 -0.60988 0.3719534 0.3719534 1 1.144687 0.1133131 0.6841737

1 1.1300496 -0.73783 0.5443935 -0.73783 0.5443936 0.5443936 1 1.144687 -0.014637 -0.088379

1 1.1338861 -0.733994 0.5387469 -0.733994 0.538747 0.538747 1 1.144687 -0.010801 -0.065215

1 1.2010994 -0.66678 0.4445962 -0.666781 0.4445963 0.4445963 1 1.144687 0.0564124 0.3406127

1 1.1008969 -0.766983 0.588263 -0.766983 0.5882631 0.5882631 1 1.144687 -0.04379 -0.264401

1 0.9756352 -0.892245 0.7961007 -0.892245 0.7961008 0.7961008 1 1.144687 -0.169052 -1.020719

1 1.0680056 -0.799874 0.6397989 -0.799874 0.639799 0.6397989 1 1.144687 -0.076681 -0.462995

1 1.0819913 -0.785889 0.6176209 -0.785889 0.617621 0.617621 1 1.144687 -0.062696 -0.378551

4 4.6410118 2.7731319 7.6902603 2.7731318 7.6902598 7.6902601 16 4.3352439 0.3057679 1.846197

4 3.769606 1.9017261 3.6165622 1.901726 3.6165619 3.6165621 16 4.3352439 -0.565638 -3.415267

4 4.2387837 2.3709038 5.6211848 2.3709037 5.6211844 5.6211846 16 4.3352439 -0.09646 -0.582417

4 4.0972028 2.2293229 4.9698807 2.2293228 4.9698803 4.9698805 16 4.3352439 -0.238041 -1.437269

4 4.4405359 2.572656 6.618559 2.5726559 6.6185586 6.6185588 16 4.3352439 0.1052921 0.6357433

4 4.2808994 2.4130194 5.8226628 2.4130194 5.8226624 5.8226626 16 4.3352439 -0.054345 -0.328127

4 4.3557894 2.4879095 6.1896937 2.4879094 6.1896933 6.1896935 16 4.3352439 0.0205456 0.1240521

4 4.5008108 2.6329309 6.9323249 2.6329308 6.9323245 6.9323247 16 4.3352439 0.1655669 0.9996769

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Document ID: Uncertainty example for THC.DOC Date Effective: 16/11/2005

Page 9 of 14

4 4.390149 2.5222691 6.3618413 2.522269 6.3618409 6.3618411 16 4.3352439 0.0549051 0.3315118

4 4.3866087 2.5187288 6.3439947 2.5187287 6.3439943 6.3439945 16 4.3352439 0.0513648 0.3101359

4 4.6894656 2.8215856 7.9613455 2.8215856 7.961345 7.9613453 16 4.3352439 0.3542217 2.1387562

Sum 55.44 61.640037 -4.44E-15 107.06105 -2.75E-06 107.06105

Average 1.68 1.8678799

n 33 33

Table 3. Summary of statistical outputs associated with calibration curves for determination of uncer tainty for 9-THC

SUMMARY OUTPUT

r= 0.995892

Regression Statistics b= 1.063519

Multiple R 0.9958922 a= 0.081168

R Square 0.9918013 Six/y 0.1682705

Adjusted R Square 0.9915368 Sb 0.017367

Standard Error 0.1682705 Sa 0.041344

Observations 33 Sxx 107.06105

ANOVA

df SS MS F Significance F

Regression 1 106.18328 106.18328 3750.0768 6.569E-34

Residual 31 0.8777638 0.028315

Total 32 107.06105

Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%

Intercept 0.0811681 0.0413437 1.9632512 0.0586422 -0.003153 0.1654892 -0.003153 0.1654892

X Variable 1 1.0635189 0.017367 61.237871 6.569E-34 1.0280987 1.0989392 1.0280987 1.0989392

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Uncertainty example for THC.DOC Page 10 of 14

To calculate relative uncertainty associated with c alibration curves:

UPredC = 2)(11

Sxx

CC

nPM

S calunknown −++

Where Sxx = �=

−

−n

j

CCj1

2)(

And where:

� UConcentration = Uncertainty associated with calibration model

� S = Std Error (residual standard deviation) = 0.1682705

� M = Slope of calibration curve = 1.0635189

� P = no of measurements taken to determine unknown = 1

� n = total number of measurements for calibration = 33

� CUnknown = determined concentration of unknown from equation = 23.425 *

� Ccal = mean value of calibration concentration = 1.68

� Sxx = 107.06105

*This assumes that if 25 ng/mL of 9-THC was added then the resultant expected (y) concentration would be 23.425ng/mL.

Hence

UPredC = 2

107.061051.68)(23.425

331

11

1.06351890.1682705 −++

= 4.41659.0311.06351890.1682705 ++

= 5.446591.06351890.1682705

UPredC = 0.369 ng/mL

UPred = 25ng/mL 0.369

= 0.01476 ng/mL

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Estimation of precision uncertainty from Quality Control Material UQC

Quality control data produced from several analysts give data on precision uncertainty (UQC) of the method.

From our QC reporter the following results have been obtained from the last 24 months of assays:

QC name Batch no. and number of

points

Target Concentration

(ng/mL)

Batch mean

(ng/mL)

CV (%)

QC THC 05/2005

n=27

20.21 19.71 7.3

QCTHCSAL 10/2004, n=18 10 10.6 8.2

QCTHCSAL 12/2004, n=20 10 10.4 4.2

QCTHCSAH 10/2004 n=19 20 21.04 6.9

QCTHCSAH 12/2004 n = 20 50 50.8 4.4

So 6.2 % is average CV over 24 months.

Uncertainty associated with mass of test sample Um

A blood sample (1 mL) weighed 10 times will enable the uncertainty to be determined for the volume of sample dispensed for analysis.

Replicate Mass of test samples (g) 1 1.08067 2 1.07679 3 1.07882 4 1.07511 5 1.07556 6 1.07313 7 1.07684 8 1.06637 9 1.07661 10 1.07366

Mean 1.075356 SD 0.0038694

The mean was and 1.075 g the SD 0.0038694.

UPipettor 1000� L = ( USpecification 1000� L)

2 + (ULab repeatability 1000� L)2

= (0.001847)2 + (0.0045)2

= (0.0000034) + (0.002025) = 0.045037 UMassSample = ( UMassSample)2

= (0.001847)2

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= (0.0000034) = 0.045037

UCombinedMTSample = ( Upipettor 1000� L)2 + (UMassSample)2

= (0.001847)2 + (0.0038694)2

= (0.0000034) + (0.0000149) = 0.00428

Combining Uncertainties

CUCombined =

2

test

m

2

QC

QC

2

GC

Pred

2

std

WorkingStd

MassU

Precision

U

nCalibratioU

Working

U��

��

�+�

��

��

�+��

�

��

�+��

�

��

�

= 2222

10003.2

1006.2

250.369

10.03

��

��

�+��

��

�+��

��

�+��

��

�

= 000010 0003844000021000090 .... +++

= 004979.0

CUCombined = 0.07 ng/mL

If C = 100 ng/mL

Uncertaintycombined = C x 0.07 ng/mL

= 100 x 0.07 ng/mL

= 7 ng/mL

Uncertaintycombined = k x UExpanded

Where k =2 for a 95% confidence interval

UExpanded = expanded uncertainty of the measure


= 2 x 7 ng/mL

= 14 ng/mL

So at 100 ng ± 14 ng

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Part II. Top Down Approach

Performance of Quality Controls

Since most of the uncertainty is assessed through quality controls [preparation of standards, variation in peak area ratio it is apposite to also use QCs to estimate uncertainty].

Validation data for intra-assay variability CV of quality controls. From our QC reporter the following results have been obtained from the last 24 months of assays:

QC name Batch no. and number of

points

Target Concentration

(ng/mL)

Batch mean

(ng/mL)

CV (%)

QC THC 05/2005

n=27

20.21

(observed)

19.71

(expected)

7.3

S=1.44

QCTHCSAL 10/2004, n=18 10 10.6 8.2

QCTHCSAL 12/2004, n=20 10 10.4 4.2

QCTHCSAH 10/2004 n=19 20 21.04 6.9

QCTHCSAH 12/2004 n = 20 50 50.8 4.4 The average CV is 6.2 % over 24 months.

Long Term precision = 6.2 %

Bias = 1.44 @ 19.71

Bias = 19.71(observed) – 20.21(expanded)

= -0.5

Where n

SUbias = 1

27

44.1=biasU

28.0=biasU

bUBias 2≤ , so bias is not significant.

22 28.044.1 +=cU

0784.00736.2 +=cU

1 O’Donnell, G & Hibbert D. Anlayst 130, 721-29(2005)

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5.1=cU

( ) 3%95 =CIU

So C = 20 ng ± 3 ng for qc data. So using the top-down approach

If concentration = 100 ng/mL, then the CV of 6.2 % can give an estimated uncertainty of

UncertaintyQC = C x 6.2%

= 6.2 ng/mL

UncertaintyQC = k x UExpanded

Where k =2 for a 95% confidence interval

UExpanded = expanded uncertainty of the measure


= 2 x 6.2 ng/mL

= 12.4 ng/mL

Dr Dimitri Gerostamoulos PhD Monash BSc(Hons)

Manager - Toxicology

Victorian Institute of Forensic Medicine

57-83 Kavanagh St., Southbank 3006

Victoria, AUSTRALIA

[email protected]

Uncertainty Example for Thc 1

Documents

uncertainty example

standard uncertainty

estimation of uncertainty

measurement uncertainty

uncertainty of dilution

uncertainty of purity

sources of uncertainty

ng of thc