8 SAMPLE PREPARATION: AN ANALYTICAL PERSPECTIVE 00000 20 Pharmaceuticals in 1E+001.E-021.E-041E-061.E-081.E-101E-12 Figure 1.4. Reproducibility as a function of concentration during analytical measurements. (Reproduced from Ref. 3 with permission from LC-GC North America. instruments have become quite sophisticated and provide high levels of accuracy and precision. On the other hand, sample preparation often re- mains a rigorous process that accounts for the majority of the variability Going back to the example of the measurement of pesticides in fish, the final analysis may be carried out in a modern computer-controlled gas chromatograph/mass spectrograph(GC-MS). At the same time, the sample preparation may involve homogenization of the liver in a grinder, followed by Soxhlett extraction, concentration, and cleanup. The sample preparation might take days, whereas the GC-Ms analysis is complete in a matter of minutes. The sample preparation also involves several discrete steps that involve manual handling. Consequently, both random and systematic errors are higher during sample preparation than during analysis The relative contribution of sample preparation depends on the steps the measurement process. For instance, typically two-thirds of the time in an analytical chromatographic procedure is spent on sample preparation. An example of the determination of olanzapine in serum by high-performance iquid chromatography/mass spectroscopy(HPLC-MS)illustrates this point 3. Here, samples were mixed with an internal standard and cleaned up in a
instruments have become quite sophisticated and provide high levels of accuracy and precision. On the other hand, sample preparation often remains a rigorous process that accounts for the majority of the variability. Going back to the example of the measurement of pesticides in fish, the final analysis may be carried out in a modern computer-controlled gas chromatograph/mass spectrograph (GC-MS). At the same time, the sample preparation may involve homogenization of the liver in a grinder, followed by Soxhlett extraction, concentration, and cleanup. The sample preparation might take days, whereas the GC-MS analysis is complete in a matter of minutes. The sample preparation also involves several discrete steps that involve manual handling. Consequently, both random and systematic errors are higher during sample preparation than during analysis. The relative contribution of sample preparation depends on the steps in the measurement process. For instance, typically two-thirds of the time in an analytical chromatographic procedure is spent on sample preparation. An example of the determination of olanzapine in serum by high-performance liquid chromatography/mass spectroscopy (HPLC-MS) illustrates this point [3]. Here, samples were mixed with an internal standard and cleaned up in a −70 −60 −50 −40 −30 −20 −10 0 10 20 30 40 50 60 70 1.E+00 1.E−02 1.E−04 1.E−06 1.E−08 1.E−10 1.E−12 Concentration Relative standard deviation Major components Minor components Trace Analysis Pharmaceuticals Drugs in feeds Pesticide residues Aflatoxins Figure 1.4. Reproducibility as a function of concentration during analytical measurements. (Reproduced from Ref. 3 with permission from LC-GC North America.) 8 sample preparation: an analytical perspective
ERRORS IN QUANTITATTVE ANALYSIS ACCURACY AND PRECISION solid-phase extraction(SPE)cartridge. The quantitation was done by a cali- bration curve. The recovery was 87 4% for three assays, whereas repeat bility of 10 replicate measurements was only I to 2%. A detailed error analysis 3 showed that 75% of the uncertainty came from the SPE step and the rest came from the analytical procedure. Of the latter, 24% was attrib- uted to uncertainty in the calibration, and the remaining 1% came from the variation in serum volume. It is also worth noting that improvement in the calibration procedure can be brought about by measures that are signifi cantly simpler than those required for improving the SPe. The variability in SPE can come from the cartridge itself, the washing, the extraction, the drying, or the redissolution steps. There are too many variables to control ome useful approaches to reducing uncertainty during sample prepara tion are given beloy Minimize the Number of steps In the example above, the sample preparation contributed 75% of the error When multiple steps such as those shown in Figure 1. 2 are involved, the 1.5 illustrates this point. A 1000-fold dilution can be performed in one step I mL to 1000 mL. It can also be performed in three steps of 1: 10 dilutions each. In the one-step dilution, the uncertainty is from the uncertainty in the volume of the pipette and the flask. In the three-step dilution, three pipettes and three flasks are involved, so the volumetric uncertainty is compounded that many times. A rigorous analysis showed [3] that the uncertainty in the one-step dilution was half of what was expected in the three-step process If and when possible, one or more sample preparation steps(Figure 1.2) should be eliminated. The greater the number of steps, the more errors there are. For example, if a cleanup step can be eliminated by choosing a selective extraction procedure, that should be adapted Use Appropriate Techniques Some techniques are known to provide higher variability than others. The choice of an appropriate method at the outset can improve precision. For example, a volume of less than 20 mL can be measured more accurately and precisely with a syringe than with a pipette. Large volumes are amenable to precise handling but result in dilution that lowers sensitivity. The goal should be to choose a combination of sample preparation and analytical instrumentation that reduces both the number of sample preparative steps nd the rSD. automated techniques with less manual handling tend to have higher precision
solid-phase extraction (SPE) cartridge. The quantitation was done by a calibration curve. The recovery was 87 G4% for three assays, whereas repeatability of 10 replicate measurements was only 1 to 2%. A detailed error analysis [3] showed that 75% of the uncertainty came from the SPE step and the rest came from the analytical procedure. Of the latter, 24% was attributed to uncertainty in the calibration, and the remaining 1% came from the variation in serum volume. It is also worth noting that improvement in the calibration procedure can be brought about by measures that are signifi- cantly simpler than those required for improving the SPE. The variability in SPE can come from the cartridge itself, the washing, the extraction, the drying, or the redissolution steps. There are too many variables to control. Some useful approaches to reducing uncertainty during sample preparation are given below. Minimize the Number of Steps In the example above, the sample preparation contributed 75% of the error. When multiple steps such as those shown in Figure 1.2 are involved, the uncertainty is compounded. A simple dilution example presented in Figure 1.5 illustrates this point. A 1000-fold dilution can be performed in one step: 1 mL to 1000 mL. It can also be performed in three steps of 1: 10 dilutions each. In the one-step dilution, the uncertainty is from the uncertainty in the volume of the pipette and the flask. In the three-step dilution, three pipettes and three flasks are involved, so the volumetric uncertainty is compounded that many times. A rigorous analysis showed [3] that the uncertainty in the one-step dilution was half of what was expected in the three-step process. If and when possible, one or more sample preparation steps (Figure 1.2) should be eliminated. The greater the number of steps, the more errors there are. For example, if a cleanup step can be eliminated by choosing a selective extraction procedure, that should be adapted. Use Appropriate Techniques Some techniques are known to provide higher variability than others. The choice of an appropriate method at the outset can improve precision. For example, a volume of less than 20 mL can be measured more accurately and precisely with a syringe than with a pipette. Large volumes are amenable to precise handling but result in dilution that lowers sensitivity. The goal should be to choose a combination of sample preparation and analytical instrumentation that reduces both the number of sample preparative steps and the RSD. Automated techniques with less manual handling tend to have higher precision. errors in quantitative analysis: accuracy and precision 9
SAMPLE PREPARATION: AN ANALYTICAL PERSPECTIVE 1000m Figure 1.5. Examples of single and multiple dilution of a sample (Reproduced from Ref. 3 with permission from LC-GC North America 1. 2.3. Statistical Aspects of Sample Preparation Uncertainty in a method can come from both the sample preparation and the analysis. The total variance is the sum of the two factors (1.4) The subscript T stands for the total variance; the subscripts s and a stand for he sample preparation and the analysis, respectively. The variance of the analytical procedure can be subtracted from the total variance to estimate the variance from the sample preparation. This could have contribution from the steps shown in Figure 1.2 C-=01+0+0+ where gh relates to homogenization, ex to extraction, o to concentration and acl to cleanup. Consequently, the overall precision is low even when
1.2.3. Statistical Aspects of Sample Preparation Uncertainty in a method can come from both the sample preparation and the analysis. The total variance is the sum of the two factors: s2 T ¼ s2 s þ s2 a ð1:4Þ The subscript T stands for the total variance; the subscripts s and a stand for the sample preparation and the analysis, respectively. The variance of the analytical procedure can be subtracted from the total variance to estimate the variance from the sample preparation. This could have contribution from the steps shown in Figure 1.2: s2 s ¼ s2 h þ s2 ex þ s2 c þ s2 cl ð1:5Þ where sh relates to homogenization, sex to extraction, sc to concentration, and scl to cleanup. Consequently, the overall precision is low even when 1 ml 1000 ml 10 ml 1 ml Figure 1.5. Examples of single and multiple dilution of a sample. (Reproduced from Ref. 3 with permission from LC-GC North America.) 10 sample preparation: an analytical perspective
ERRORS IN QUANTITATIVE ANALYSIS: ACCURACY AND PRECISION II a high-precision analytical instrument is used in conjunction with low precision sample preparation methods. The total variance can be estimated by repeating the steps of sample preparation and analysis several times. Usually, the goal is to minimize the number of samples, yet meet a spe- cific level of statistical certainty. The total uncertainty, E, at a specific con- fidence level is selected. The value of e and the confidence limits are deter mined by the measurement quality required E (1.6) where a is the standard deviation of the measurement, z the percentile of standard normal distribution, depending on the level of confidence, and n the number of measurements. If the variance due to sample preparation, as is negligible and most of the uncertainty is attributed to the analysis, the minimum number of analysis per sample is given by ea (1.7) The number of analyses can be reduced by choosing an alternative method with higher precision (i.e, a lower a)or by using a lower value of = which means accepting a higher level of error. If the analytical uncertainty negligible(oa-0) and sample preparation is the major issue, the minimum number of samples, ns, is given by (1.8) Again, the number of samples can be reduced by accepting a higher uncer- tainty or by reducing s. When a and os are both significant, the total error Er is given by E (1.9) This equation does not have an unique solution. The same value of error Er, can be obtained by using different combinations of ns and na Comb nations of ns and na should be chosen based on scientific judgment and the cost involved in sample preparation and analysis
a high-precision analytical instrument is used in conjunction with lowprecision sample preparation methods. The total variance can be estimated by repeating the steps of sample preparation and analysis several times. Usually, the goal is to minimize the number of samples, yet meet a specific level of statistical certainty. The total uncertainty, E, at a specific con- fidence level is selected. The value of E and the confidence limits are determined by the measurement quality required: E ¼ zs ffiffiffi n p ð1:6Þ where s is the standard deviation of the measurement, z the percentile of standard normal distribution, depending on the level of confidence, and n the number of measurements. If the variance due to sample preparation, s2 s , is negligible and most of the uncertainty is attributed to the analysis, the minimum number of analysis per sample is given by na ¼ zsa Ea 2 ð1:7Þ The number of analyses can be reduced by choosing an alternative method with higher precision (i.e., a lower sa) or by using a lower value of z, which means accepting a higher level of error. If the analytical uncertainty is negligible ðsa ! 0Þ and sample preparation is the major issue, the minimum number of samples, ns, is given by ns ¼ zss Es 2 ð1:8Þ Again, the number of samples can be reduced by accepting a higher uncertainty or by reducing ss. When sa and ss are both significant, the total error ET is given by ET ¼ z s2 s ns þ s2 a na 1=2 ð1:9Þ This equation does not have an unique solution. The same value of error, ET , can be obtained by using di¤erent combinations of ns and na. Combinations of ns and na should be chosen based on scientific judgment and the cost involved in sample preparation and analysis. errors in quantitative analysis: accuracy and precision 11
SAMPLE PREPARATION: AN ANALYTICAL PERSPECTIVE A simple approach to estimating the number of samples is to repeat the sample preparation and analysis to calculate an overall standard deviation, Using Students t distribution, the number of samples required to achieve a given confidence level is calculated as (1.10) where t is the f-statistic value selected for a given confidence level and e is he acceptable level of error. The degrees of freedom that determine t can first be chosen arbitrarily and then modified by successive iterations until the number chosen matches the number calculated Relative standard deviation of repeat HPLC analysis of a drug metabolite standard was between 2 and 5%. Preliminary measurements of several seru samples via solid-phase extraction cleanup followed by HPLC analyses showed that the analyte concentration was between 5 and 15 mg/L and the standard deviation was 2.5 mg/L. The extraction step clearly increased he random error of the overall process. Calculate the number of samples required so that the sample mean would be within +1.2 mg/L of the popu lation mean at the 95% confidence level Using equation(1.10), assuming 10 degrees of freedom, and referring to the t-distribution table from a statistics textbook, we have t= 2.23, 5=2.5 and e= 1. 2 mg/L, so n=(2.23 x 2.5/1. 2)-=21.58 or 22. Since 22 is sig nificantly larger than 10, a correction must be made with the new value of t corresponding to 21 degrees of freedom([= 2.08): n=(2.08 x 2.5/1.2) 18.78 or 19. Since 19 and 22 are relatively close, approximately that many samples should be tested. A higher level of error, or a lower confidence level, may be accepted for the reduction in the number of samples 1.3. METHOD PERFORMANCE AND METHOD VALIDATION The criteria used for evaluating analytical methods are called figures of merit. Based on these characteristics, one can predict whether a method meets the needs of a certain application. The figures of merit are listed in Table 1. 2. Accuracy and precision have already been discussed; other im- portant characteristics are sensitivity, detection limits, and the range of
A simple approach to estimating the number of samples is to repeat the sample preparation and analysis to calculate an overall standard deviation, s. Using Student’s t distribution, the number of samples required to achieve a given confidence level is calculated as n ¼ ts e 2 ð1:10Þ where t is the t-statistic value selected for a given confidence level and e is the acceptable level of error. The degrees of freedom that determine t can first be chosen arbitrarily and then modified by successive iterations until the number chosen matches the number calculated. Example Relative standard deviation of repeat HPLC analysis of a drug metabolite standard was between 2 and 5%. Preliminary measurements of several serum samples via solid-phase extraction cleanup followed by HPLC analyses showed that the analyte concentration was between 5 and 15 mg/L and the standard deviation was 2.5 mg/L. The extraction step clearly increased the random error of the overall process. Calculate the number of samples required so that the sample mean would be within G1.2 mg/L of the population mean at the 95% confidence level. Using equation (1.10), assuming 10 degrees of freedom, and referring to the t-distribution table from a statistics textbook, we have t ¼ 2:23, s ¼ 2:5, and e ¼ 1:2 mg/L, so n ¼ ð2:23 2:5=1:2Þ 2 ¼ 21:58 or 22. Since 22 is significantly larger than 10, a correction must be made with the new value of t corresponding to 21 degrees of freedom ðt ¼ 2:08Þ: n ¼ ð2:08 2:5=1:2Þ 2 ¼ 18:78 or 19. Since 19 and 22 are relatively close, approximately that many samples should be tested. A higher level of error, or a lower confidence level, may be accepted for the reduction in the number of samples. 1.3. METHOD PERFORMANCE AND METHOD VALIDATION The criteria used for evaluating analytical methods are called figures of merit. Based on these characteristics, one can predict whether a method meets the needs of a certain application. The figures of merit are listed in Table 1.2. Accuracy and precision have already been discussed; other important characteristics are sensitivity, detection limits, and the range of quantitation. 12 sample preparation: an analytical perspective