Talanta 48 (1999) 729736

Short communication

Intra-laboratory testing of method accuracy from recoveryassays

A. Gustavo Gonzalez *, M. Angeles Herrador, Agustn G. AsueroDepartment of Analytical Chemistry, Uni6ersity of Se6ille, 41012, Se6ille, Spain

Received 27 April 1998; received in revised form 6 August 1998; accepted 7 August 1998

Abstract

A revision on intra-laboratory testing of accuracy of analytical methods from recovery assays is given. Proceduresbased on spiked matrices and spiked samples are presented and discussed. 1999 Elsevier Science B.V. All rightsreserved.

Keywords: Accuracy; Recovery assays; Spiked samples; Spiked matrices

1. Introduction

The accuracy of an analytical method is a keyfeature for validation purposes [1,2]. Four princi-pal methods have been proposed to the study ofaccuracy of analytical methods [35]. They arebased on: (i) the use of certified reference materi-als (CRM); (ii) the comparison of the proposedmethod with a reference one, (iii) the use ofrecovery assays on matrices or samples and, (iv)the round robin studies (collaborative tests).

CRMs, when available, are the preferred con-trol materials because they are directly traceableto international standards or units. The procedureconsists in analyzing a sufficient number of CRMs

and comparing the results against the certifiedvalues [4,6,7]. The Community Bureau of Refer-ence of the Commission of the European Commu-nity (BCR, Bureau Communautaire deReference), the Laboratory of the GovernmentChemist of Middlesex (LGC), the National Insti-tute of Standards and Technology of USA (NIST)and the National Institute for EnvironmentalStudies of Japan, provide a general coverage ofcertified reference materials [810]. Nevertheless aseries of shortcomings and limitations for CRMshave been pinpointed [11], specially: Their cost,the small amounts that may be purchased and thenarrow range covered of matrices and analytes.

The performance of a newly developed methodcan be assessed by comparing the results obtainedby it with those found with a reference or com-parison method of known accuracy and precision[1216].

* Corresponding author. Tel.: 34 5 4557173; fax: 34 54557168; e-mail: agonzale@cica.es

0039-9140:99:$ - see front matter 1999 Elsevier Science B.V. All rights reserved.

PII S0039-9140(98)00271-9

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The use of Collaborative studies to controlmethodological bias is a very important topic[17,18]. However proficiency testing and roundrobin studies will not be considered here, thispaper dealing with the internal (intra-labora-tory) control of accuracy.

Unfortunately, within the realm of environ-mental, toxicological and pharmaceutical analy-sis, neither CRMs nor alternate methods areavailable for new contaminant, toxic and drug-related analytes. Accordingly, the remainingmethods to check method accuracy, that is, therecovery assays are the tools of the trade forthe study of accuracy in pharmaceutical analy-sis.

The aim of the present paper is to revise,outline and discuss the suitable procedures fortesting accuracy in pharmaceutical analysis fromrecovery assays. For the sake of illustrationsome examples taken from literature are dis-cussed.

2. General overview on recovery assays

Before embarking us in the body of the sub-ject, some terminology should be established.According to the IUPAC paper (1990) onnomenclature for sampling in analytical chem-istry [19,20] we will use the term test portion asthe quantity of material removed from the testsample which is suitable in size to measure theconcentration of the determinant by using theselected analytical method. The test portionmay be dissolved with or without reaction togive the test solution. An aliquot (fractionalpart) of the test solution is then measured byfollowing the analytical operating procedure.The test portion consists of analyte plus matrix[21].

Accordingly, if a test portion of weight m isdissolved into a total volume V, the test solu-tion will have a concentration cm:V which isthe sum of the analyte concentration (x) plusthe matrix concentration (z).

Consider now the newly proposed analyticalmethod which is applied to dissolved test por-tions of a given sample within the linear dy-

namic range of the analytical response (Y).This response may be expressed by the follow-ing relationship involving both analyte and ma-trix amounts [22]:

YABxCzDxz (1)

where A, B, C and D are constantsA is a constant that does not change when

the concentrations of the matrix, z, and:or theanalyte, x, change. It is called the true sampleblank [23] and may be evaluated by using theYouden sample plot [2426], which is definedas the sample response curve [23]. By usingour terminology, the application of the selectedanalytical operating procedure to different testportions, m, (different mass taken from the testsample) produces different analytical responsesY as outputs. The plot of Y versus m is theYouden sample plot and the intercept of thecorresponding regression line is the so calledTotal Youden Blank (TYB) which is the truesample blank [2330]. As will be discussed be-low, when a matrix without analyte is avail-able the term A can be more easily determined.

Bx is the fundamental term that justifies theanalytical method and it is directly related tothe analytical sensitivity [31].

Cz is the contribution from the matrix, de-pending only on its amount, z. When this termoccurs, the matrix is called interferent. In gen-eral this kind of interference is very infrequent,because a validated analytical method should beselective enough with respect to the potentialinterferences appearing in the samples wherethe analyte is determined [32]. The USP mono-graph [33] defines the selectivity of an analyticalmethod as its ability to measure accurately ananalyte in presence of interference, such as syn-thetic precursors, excipients, enantiomers andknown or likely degradation products that maybe present in the sample matrix. Accordingly,the majority of validated methods do not sufferfrom such a direct matrix interference. Anyway,as it will be discussed below, the method accu-racy may be tested even when faced with inter-ferent matrices.

Dxz is an interaction analyte:matrix term.This matrix effect occurs when the sensitivity of

A.G. Gonzalez et al. : Talanta 48 (1999) 729736 731

the instrument to the analyte is dependent on thepresence of the other species (the matrix) in thesample [31]. For the sake of determining analytes,this effect may be overcome by using the methodof standard additions (MOSA) [2330].

Certain types of samples, of which pharmaceu-tical dosage forms are just an example, enablesthe sample matrix to be simulated by a laboratorypreparation procedure with all the excipientspresent in their corresponding amounts except theanalyte of interest. In other cases, the samplematrix may be a synthetic mixture of naturallycomplex substances such as an analyte-free bodyfluids from unmedicated patients. In both cases, itis said that the placebo is available and recoveriesare obtained from spiked placebos. The termplacebo was used by Cardone [28] to refer tofree-analyte materials. However in order to avoidconfusion and unify the jargon, the word matrix(or blank matrix) will be used throughout thetext. Sometimes, however, it is not possible toprepare a matrix without the presence of analyte.This may occur, for instance with lyophilizedmaterials, in which the speciation is significantlydifferent when the analyte is absent [3]. In thesecases, the matrix is not available and the MOSAmust be applied [2330] the recoveries being ob-tained from spiked samples. In spiked matrices orspiked samples, the analyte addition cannot beblindly performed. Addition should be accom-plished over the suitable analyte range.

Some analytes when incorporated naturally intothe matrix are chemically bound to the con-stituents of the matrix. In such cases, the mereaddition to the sample or matrix will not mirrorwhat happens in practice. It is recommended thatthe analyte is added to the matrix and then left incontact for several hours, preferably overnight,before applying the analytical method to allowanalyte:matrix interactions to occur [34]. Spikedsamples or matrices are also called fortified ones,the concentration of analyte added being the cor-responding fortification level. In the followingthe two procedures for demonstrating accuracyfrom recovery tests, namely, spiked matrices andspiked samples will be outlined. For the sake ofgenerality, in all cases the coefficient C of Eq. (1)will be considered significant.

3. Recovery tests from spiked matrices

The recovery test is carried out from spikedmatrices over the analyte range of interest, gener-ally 75125% of the expected assay value (labelclaim or theory), holding the matrix at the nomi-nal constant level zz0. The analytical responsewhen analyte is added follows the equation

YABxCz0Dxz0

(ACz0) (BDz0)xA %B %x (2)

where A % and B % are the intercept and the slope ofthe calibration line in the matrix environmentwhich are constant because B, C, D and z0 arealso fixed.

By using this calibration function, severalamounts of analyte are added to the matrix. Fromthe analytical signal at each addition i, Yi, theamount of analyte found is estimated as xi(YiA %):B %. The recovery (Rec) may be estimatedas the average of the individual recoveries ob-tained at each spike i (Reci xi:xi). Alternatively,a regression analysis of found versus added ana-lyte concentrations may be performed, and theslope may be taken as the average recovery as itwill be discussed in Section 5.

4. Recovery tests from spiked samples

In a way similar to the preceding section, theapplication of the MOSA on the test portions(matrix plus analyte) is performed. An importantrequirement for this technique is that all solu-tions, unspiked and spiked test portions be dilutedto the same final volume. If we carry out analyteadditions, x, on the final solution of a given testportion of concentration c0x0z0 (x0 is theconcentration of analyte coming from the sample,present in the final solution), then the analyticalresponse will be:

YAB(x0x)Cz0D(x0x)z0

(ACz0) (BDz0)(x0x)

A %B %(x0x)

A %B %x0B %xAB %x (3)

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Note that both A and B % are constant because A,B, C, D, x0 and z0 are also constants. By usingthis calibration function, from the analytical sig-nal at each addition i, Yi, the amount of analytefound is estimated as xi (YiA):B %. This is anestimation of the added analyte concentration (xi)instead of the total analyte concentration (x0xi). A crucial requirement is that, the total finalanalyte concentration obtained for the maximumamount of analyte spiked should remain withinthe linear range obtained at the method develop-ment step. The remainder is the same as it wasdiscussed above.

5. Evaluation of the recovery from spikedmatrices or samples and significance tests forassessing accuracy

The calculation of recoveries from spiked ma-trices or samples may be performed (i) by com-puting the individual recoveries for each spikedamount (fortification level) of analyte, and evalu-ating the recovery as average; and (ii) from linearregression analysis of added versus found data.

5.1. Method of the a6eraged reco6ery

In this case, for each fortification level or ana-lyte spike i, we have the concentration added ofanalyte xi. From the calibration graph, the esti-mated concentration of analyte xi is obtained. Anindividual recovery is then calculated as Reci xi:xi. The mean recovery, Rec, is calculated as aver-age of the individual ones

Rec1n

%in

i1

Reci (4)

The average recovery may be tested for signifi-cance by using the Student t-test, the null hypoth-esis being that the recovery is unity (or 100 inpercentage) and the method is accurate.

tRec1

sRec:n(5)

with

sRecD%ini1 (RecReci)2

n1(6)

n being the number of spike levels.If the t value obtained from Eq. (5) is less than

the tabulated value for n1 degrees of freedomat a given significance level, then the null hypoth-esis is accepted and the method is accurate.

5.2. Regression analysis of added 6ersus founddata

Both in spiked matrices or samples, a regressionanalysis of estimated (found) against spiked(added) analyte concentration is performed. Thesestudies are not new, indeed. In a landmark paper[35], Mandel and Linnig studied the accuracy inchemical analysis using linear calibration curves,by applying regression analysis to the linearrelationships

xfoundabxadded (7)

here xfound and xadded refer to the above concen-trations of analyte estimated (x) and spiked (x).The theory predicts a value of 1 for the slope, b,and a value of 0 for the intercept. However, theoccurrence of systematic and random errors in theanalytical procedure may produce deviations ofthe ideal situation [36]. Thus, it may occur thatthe straight line has a slope of 1 but a non-zerointercept, coming from a wrongly estimated back-ground signal, and reflecting the need for a blankcorrection in the calibration graph. Another pos-sibility is that the slope is significantly differentfrom unity, indicating a source of proportionalerror in the proposed analytical method. The plotmay present curvature or even may exhibit pecu-liar behaviour in case of analyte speciation.

Once the parameters a and b were calculatedfrom the linear fit xfound versus xadded, and beforeto evaluate the recovery, diagnostic checking ofthe residuals (responsesmodel predictions), thatis, xfoundabxadded at each spike level, shouldbe applied to assess model fit validity [37]. Certainunderlying assumption have been outlined for theregression analysis such as the independence ofthe random error, homoscedasticity and Gaussian

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distribution of the random error [38]. If the modelrepresents the data suitably, the residuals shouldbe randomly distributed about the value predictedby the model equation with normal distribution. Aplot of the residuals on a normal probability paperis a useful technique [39]. If the error distributionis normal, then the plot will be linear. On the otherhand, examinations of plots of residuals againstthe independent variable (here xadded) may be ofgreat help in the diagnosis of the regression models[40]. Some systematic patterns indicated that themodel is incorrect in some way. A sector patternindicates heteroscedasticity in the data [41,42]. Anon linear pattern indicates that the present modelis incorrect [43]. Residuals also may be used todetect outliers. A very straightforward way is toconsider as outlier any calibration point whoseresidual is greater than twice the value of thestandard deviation of the regression line, althoughthe use of jackknife residuals or the Cook distancemethod are more accurate tools for detectingoutliers [44].

The heteroscedasticity revealed by residual anal-ysis of the added versus found data comes from theheteroscedasticity in the responses (Y) of the cali-bration curve which is propagated to the estimatesxfound. In such a case, instead of using ordinaryleast squares methods to fit the calibration straightline, the use of weighted least squares is advised.The weights wi are given by wi1:s i2, where si isthe standard deviation of the responses, replicatedat the concentration of analyte xi [41,42,45].

Non-linear patterns detected by residual analy-sis of the added versus found plots arise becausethe calibration graph used for estimating the ana-lyte concentration is by far non-linear. Curvedpatterns suggest the inclusion of a quadratic termin x2 in the calibration function. In these situa-tions, suitable non-linear calibration curves shouldbe established in order to obtain unbiased esti-mates of xfound. Three methods are available forthis purpose, namely the method of the linearsegments, the method of the three-parameter func-tion and the method based on polynomial func-tions [46,47].

After the validity of the fit has been appraised,then statistical comparison of a and b with theiridealistic values, 0 and 1, must be performed.

Conventional individual confidence intervals forthe slope and the intercept once their standarddeviations sb and sa are calculated, based on thet-test (ta a :sa ; tb b1:sb), although fre-quently used by the workers [4850], can lead toerroneous conclusions because these tests, whencarried out independently of each other ignore thestrong correlation between slope and intercept[51]. Instead of these individual tests, the ellipticjoint confidence region (EJCR) for the true slope(b) and intercept (a) derived by Working andHotelling [52] and adopted by Mandel and Linnig[35] is recommended, whose equation is

n(aa)22% xi(aa)(bb)

% x2i (bb)22s2F2, n (8)

where n is the number of points, s2 the regressionvariance and F2, n the critical value of theSnedecorFishers statistic with 2 and nn2degrees of freedom at a given P% confidence level,usually 95% [53,54].

The centre of ellipse is (a, b). Any point (a, b)which lies inside the EJCR is compatible with thedata at the chosen confidence level P. In order tocheck constant (translational) or proportional (ro-tational) bias, the values a0 and b1 arecompared with the estimates a and b using theEJCR. If the point (0, 1) lies inside the EJCR, thenbias are absent [13] and consequently, the recoverymay be taken as unity (or 100% in percentile scale).This can be done from easy calculations as de-scribed in Appendix A.

Once the recovery is computed (from one oranother procedure), it should be checked for fulfi-lling the accuracy criteria according to the AOACguidelines [55] as indicated in Table 1. Note thatfor trace analysis, e.g. drug residues in tissues,recoveries about 50% is often the best that can beachieved.

6. Worked example

For the sake of illustration, a case study wasselected. It deals with the recovery of trigonellinein coffee extracts from ion chromatography.

A.G. Gonzalez et al. : Talanta 48 (1999) 729736734

Trigonelline is determined in green and roastedcoffee extracts by an ion chromatography pro-cedure using as stationary phase polybutadi-enemaleic acid (PBDMA) coated on silica, 2mmol l1 aqueous hydrochloric acid (pH 3) aseluent and UV detection at 254 nm. Test solu-tions were prepared by refluxing test portionsof 3 g of dried coffee (green or roasted) withhot water (80C) for 1 h. The extract wasfiltered and diluted to 250 ml. For trigonellineanalysis, the test solution is diluted 1:5 (V:V)and 3 ml of the resulting solution is passedthrough a C18 SPE cartridge and then, the elu-ent is also passed to collect a total volume of10 ml. An aliquot of this later solution wasfiltered through a 0.45 mm filter unit and subse-quently injected on the HPLC system [56].

Owing to the lack of suitable certified refer-ence materials, the validation methodology wasbased on recovery assays from spiked extractsof green and roasted coffees. Roasted coffeeshave trigonelline contents within 0.500.85%(w:w, dry basis). The corresponding extractswill present trigonelline concentrations in therange 60102 mg l1, which corresponds to1525.5 mg trigonelline. Six spikes, additionsor fortification levels were selected. Spiked ex-tracts were allowed overnight and then ana-lyzed. The added and found amounts oftrigonelline are shown in Table 2.

The individual recoveries are also indicated inTable 2 at each fortification level. The averagedrecovery is 0.9997 and its standard deviation,0.0094. By applying Eq. (5), the observed tvalue is 0.078. The critical value for 5 degreesof freedom at a 95% confidence level is 2.015,

Table 2Recovery study of trigonelline on coffee extracts

ResidualEstimated xfoundxadded xfound Rec

0.145 5.02 1.004 4.880.059.940.98910 9.89

14.94 15.00 0.0615 0.9960.2520.0620 19.81 0.990

25.1225.30 0.181.012251.007 30.18 0.0430.2230

xadded and xfound refer to the amounts (in mg) of trigonellineadded and found by analysis in the different extracts.

and therefore, the null hypothesis is acceptedand the recovery do not differ statistically from100%. The same conclusion could be drawnfrom comparison of the averaged recovery inpercentile scale, 99.97, with the ranges providedby Table 1. Considering that trigonelline con-tents in coffee are of 0.50.85%, the third rowof the table (\0.1%) is selected and the corre-sponding recovery range is 95105%. The aver-aged recovery 99.97 lies within this interval andconsequently, meets with the requirements ofthe AOAC guidelines.

On the other hand, the regression approachcan be carried out. The plot of xadded versusxfound was linear with an intercept of 0.18and a slope of 1.012. The correlation coefficientwas about 0.9999 and the regression variances20.0355. In Table 2 the estimated values ofxfound and the corresponding residuals are pre-sented. Residual analysis did not show anypathology with the exception of a large value(0.25) for the point corresponding to the ad-dition of 20 mg. However, the absolute valueof this residual is lesser than twice the standarddeviation of the regression line (s0.1884) andhence cannot be considered as outlier. A plotof the residuals on a normal probability paperwas fairly linear, which ratifies the model valid-ity.

In order to test if simultaneously the slopeand intercept are not different from the idealis-tic values a0 and b1, the procedure baseon the EJCR was considered as it is explained

Table 1Analyte recovery depending on the concentration range

Analyte concentration (%) Recovery range (%)

]10 9810297103]1

]0.1 9510590107]0.01

]0.001--]0.00001 8011060115]0.000001

]0.0000001 40120

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in Appendix A. Thus, by taking the critical valuefor the SnedecorFisher statistic at a 95% confi-dence level F2, 46.94, we obtain b10.2304B1,and b21.7770\1. This indicates that the point(0, 1) lies inside the EJCR and then, the interceptmay be considered to be zero and the slope to beunity, which leads to that the recovery can beconsidered as 100%.

Appendix A

The equation of the isoprobability ellipse isgiven by (8). One very easy way to determine ifthe point (0, 1), corresponding to the joint nullhypothesis a0 and b1, lies inside the ellipse(and therefore the null hypothesis is accepted) isto consider the intersections of the straight linea0 and the ellipse: Only when the point (0, 1)lies inside the ellipse, the straight line a0 andthe ellipse intersect at two points (0, b1) and(0, b2) fulfilling that (for b2\b1): b2\1 andb1B1 simultaneously. Otherwise, the point (0, 1)is outside the ellipse.

If in Eq. (8) we set a0 and zbb, thefollowing expression is obtained:% x2i nz22a % xinz [na22s2F2, n ]0

(9)

after the following changes

L% x2i

M2a % xi

Nna22s2F2, n (10)

the roots for z will be

z1MM24LN

2L

z2MM24LN

2L(11)

and consequently, b1bz1 and b2bz2,which are the parameters needed to check if thepoint (0, 1) lies inside or outside the EJCR.

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