81 ANTIBIOTICS—MICROBIAL ASSAYS
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Introduction and General Information The activity (potency) of antibiotics can be demonstrated by their inhibitory effect on microorganisms under suitable conditions. A reduction in antimicrobial activity may not be adequately demonstrated by chemical methods. This chapter summarizes procedures for the antibiotics recognized in the United States Pharmacopeia (USP) for which the microbiological assay is the standard analytical method.
Two general techniques are employed: the cylinderplate (or plate) assay and the turbidimetric (or tube) assay. Table 1 lists all the antibiotics that contain microbial assays and specifies the type of assay (cylinderplate or turbidimetric).
Table 1
[Note—Perform all procedures under conditions designed to avoid extrinsic microbial contamination. Take adequate safety precautions while performing these assays because of possible allergies to drugs and because live cultures of organisms are used in the procedures. ]
Cylinderplate assay:
The cylinderplate assay depends on diffusion of the antibiotic from a vertical cylinder through a solidified agar layer in a Petri dish or plate. The growth of the specific microorganisms inoculated into the agar is prevented in a circular area or zone around the cylinder containing the solution of the antibiotic.
Turbidimetric assay:
The turbidimetric assay depends on the inhibition of growth of a microorganism in a uniform solution of the antibiotic in a fluid medium that is favorable to the growth of the microorganism in the absence of the antibiotic.
Units and Reference Standards:
The potency of antibiotics is designated in either units (U) or µg of activity. In each case the unit or µg of antibiotic activity was originally established against a United States Federal Master Standard for that antibiotic. The corresponding USP Reference Standard is calibrated in terms of the master standard.
Originally, an antibiotic selected as a reference standard was thought to consist entirely of a single chemical entity and was therefore assigned a potency of 1000 µg/mg. In several such instances, as the manufacturing and purification methods for particular antibiotics became more advanced, antibiotics containing more than 1000 µg of activity/mg became possible. Such antibiotics had an activity equivalent to a given number of µg of the original reference standard. In most instances, however, the µg of activity is exactly equivalent numerically to the µg (weight) of the pure substance. In some cases, such as those listed below, the µg of activity defined in terms of the original master standard is equal to a unit:
Do not assume that the µg of activity corresponds to the µg (weight) of the antibiotic substance.
Apparatus:
Labware used for the storage and transfer of test dilutions and microorganisms must be sterile and free of interfering residues (see Cleaning Glass Apparatus 1051). Use a validated sterilization method, such as dry heat, steam, or radiation; or use sterile, disposable labware.
Temperature control:
Thermostatic control is required in several stages of a microbial assay: when culturing a microorganism and preparing its inoculum, and during incubation in plate and tube assays. Refer to specific temperature requirements below for each type of assay.
Test organisms:
The test organism for each antibiotic is listed in Table 3 for the cylinderplate assay and Table 8 for the turbidimetric assay. The test organisms are specified by the American Type Culture Collection (ATCC) number.
In order to ensure acceptable performance of test organisms, store and maintain them properly. Establish the specific storage conditions during method validation or verification. Discard cultures if a change in the organism's characteristics is observed.
Prolonged storage:
For prolonged storage, maintain test organisms in a suitable storage solution such as 50% fetal calf serum in broth, 10%–15% glycerol in tryptic soy broth, defribinated sheep blood, or skim milk. Prolongedstorage cultures are best stored in the freezedried state; temperatures of 60 or below are preferred; temperatures below 20 are acceptable.
Primary cultures:
Prepare primary cultures by transferring test organisms from prolongedstorage vials onto appropriate media, and incubate under appropriate growth conditions. Store primary cultures at the appropriate temperature, usually 2–8, and discard after three weeks. A single primary culture can be used to prepare working cultures only for as many as seven days.
Working cultures:
Prepare working cultures by transferring the primary culture onto appropriate solid media to obtain isolated colonies. Incubate working cultures under appropriate conditions to obtain satisfactory growth for preparation of test inocula. Prepare fresh working cultures for each test day.
Uncharacteristic growth or performance of a test organism:
Use new stock cultures, primary cultures, or working cultures when a test organism shows uncharacteristic growth or performance.
Assay designs:
Suitable experimental designs are key to increasing precision and minimizing bias. Control of the incubation parameters, temperature distribution and time, is critical for minimizing bias; it can be accomplished by staging the plates and racks as described for each assay.
Cylinderplate assay:
The comparisons are restricted to relationships between zone diameter measurements within plates, excluding the variation between plates. Individual plate responses are normalized on the basis of the relative zone size of the standard compared to the mean zone size of the standard across all plates.
Turbidimetric assay:
To avoid systematic bias, place replicate tubes randomly in separate racks so that each rack contains one complete set of treatments. The purpose of this configuration is to minimize the influence of temperature distribution on the replicate samples. The turbidimetric assay, because of the configuration of the samples in test tube racks, is sensitive to slight variations in temperature. The influence of temperature variation can also be decreased by ensuring proper airflow or heat convection during incubation. At least three tubes for each sample and standard concentration (one complete set of samples) should be placed in a single rack. The comparisons are restricted to relationships between the observed turbidities within racks.
Potency considerations:
Within the restrictions listed above, the recommended assay design employs a fiveconcentration standard curve and a single concentration of each sample preparation.
For the cylinderplate assay, each plate includes only two treatments, the reference treatment (median level standard, i.e., S3) and one of the other four concentrations of the standard (S1, S2, S4, and S5) or the sample (U3). The concentration of the sample is an estimate based on the target concentration. The sample should be diluted to give a nominal concentration that is estimated to be equivalent to the median reference concentration (S3) of the standard. The purpose of diluting to the median reference concentration is to ensure that the sample result will fall within the linear portion of the standard curve. The test determines the relative potency of U3 against the standard curve. The sample (U3) should have a relative potency of about 100%. The final potency of the sample is obtained by multiplying the U3 result by the dilution factor.
An assay should be considered preliminary if the computed potency value of the sample is less than 80% or more than 125%. In this case, the results suggest that the sample concentration assumed during preparation of the sample stock solution was not correct. In such a case, one can adjust the assumed potency of the sample on the basis of the preliminary potency value and repeat the assay. Otherwise, the potency will be derived from a portion of the curve where the standard and sample responses will likely not be parallel.
Microbial determinations of potency are subject to interassay as well as intraassay variables; therefore two or more independent assays are required for a reliable estimate of the potency of a given sample. Starting with separately prepared stock solutions and test dilutions of both the standard and the sample, perform additional assays of a given sample on a different day. The mean potency should include the results from all the valid independent assays. The number of assays required in order to achieve a reliable estimate of potency depends on the variability of the assay and the required maximum uncertainty for the potency estimate. The latter is assessed by the width of the confidence interval (refer to Calculations, Confidence limits and combinations of assay calculations). The combined result of a series of smaller, independent assays spread over a number of days is a more reliable estimate of potency than one from a single large assay with the same total number of plates or tubes. Note that additional assays or lower variability allows the product to meet tighter specification ranges. Reducing assay variability achieves the required confidence limit with fewer assays.
CylinderPlate Method
Temperature control:
Use appropriately qualified and calibrated equipment to obtain the temperature ranges specified in Table 3.
Apparatus
Plates:
Glass or disposable plastic Petri dishes (approximately 20 × 100 mm or other appropriate dimensions) with lids
Cylinders:
Stainless steel or porcelain cylinders; 8 ± 0.1mm o.d.; 6 ± 0.1mm i.d.; 10 ± 0.1mm high. [Note—Carefully clean cylinders to remove all residues; occasional cleaning in an acid bath, e.g., with about 2 N nitric acid or with chromic acid (see Cleaning Glass Apparatus 1051) is required. ]
Standard solutions:
To prepare a stock solution, dissolve a suitable quantity of the USP Reference Standard of a given antibiotic, or the entire contents of a vial of USP Reference Standard, where appropriate, in the solvent specified in Table 2; and dilute to the specified concentration. Store at 2–8, and use within the period indicated. On the day of the assay, prepare from the stock solution five or more test dilutions, in which the successive solutions increase stepwise in concentration, usually in the ratio of 1:1.25. Use the final diluent specified such that the median has the concentration suggested in Table 2.
Sample solutions:
Assign an assumed potency per unit weight or volume to the sample. On the day of the assay prepare a stock solution in the same manner specified for the USP Reference Standard (Table 2). Dilute the sample stock solution in the specified final diluent to obtain a nominal concentration equal to the median concentration of the standard (S3).
Table 2
Inocula:
Suspend the test organism from a freshly grown slant or culture in 3 mL of sterile saline TS. Glass beads can be used to facilitate the suspension. Spread the saline suspension onto the surface of two or more agar plates (covering the entire surface) or onto the surface of a Roux bottle containing 250 mL of the specified medium (see Table 3).
Incubate for the specified time and at the temperature as specified in Table 3, or until growth is apparent.
After incubation, harvest the organism from the plates or Roux bottle with approximately 50 mL of sterile saline TS (except use Medium 34 for bleomycin; see the section Media and Solutions), using a sterile bent glass rod or sterile glass beads. Pipet the suspension into a sterile glass container. This is the harvest suspension.
Dilute an appropriate amount of the harvest suspension with sterile saline TS. Using the UVvisible spectrophotometer, measure % transmittance at 580 nm. The target value is approximately 25% transmittance at 580 nm. This value is used to standardize the harvest suspension volume added to the seed layer agar.
Starting with the suggested volumes indicated in Table 3, determine during method verification the proportions of stock suspension to be added to the inoculum medium that result in satisfactory zones of inhibition of approximately 14–16 mm in diameter for the median concentration of the standard (S3). [Note—Zone sizes that are outside the 11 to 19mm range are not desirable, because these contribute to assay variability. ] If the dilution percentage transmittance is above 25%, a ratio may be used to normalize the addition of organism to the seed layer. The normalization factor can be determined by dividing the percentage transmittance obtained from the dilution by 25. This ratio can then be multiplied by the suggested inoculum amount to obtain the volume (mL) of harvest suspension that needs to be added to the seed layer. Adjust the quantity of inoculum on a daily basis, if necessary, to obtain an optimum concentration–response relationship.
Alternatively, determine during method verification the proportion of harvest suspension to be incorporated into the inoculum, starting with the volumes indicated in Table 3, that result in satisfactory demarcation of the zones of inhibition of about 14–16 mm in diameter for the median concentration of the standard (S3) and giving a reproducible concentration–response relationship. Prepare the inoculum by adding a portion of stock suspension to a sufficient amount of agar medium that has been melted and cooled to 45–50. Swirl the mixture without creating bubbles in order to obtain a homogeneous suspension.
Table 3
Analysis:
Prepare the base layer for the required number of assay Petri plates, using the medium and volume shown in Table 4. Allow it to harden into a smooth base layer of uniform depth. Prepare the appropriate amount of seed layer inoculum (Table 5) as directed for the given antibiotic (Table 3) with any adjustments made based on the preparatory trial analysis. Tilt the plate back and forth to spread the inoculum evenly over the base layer surface, and allow it to harden.
Table 4 (base layer)
Table 5 (seed layer)
Drop six assay cylinders on the inoculated surface from a height of 12 mm, using a mechanical guide or other device to ensure even spacing on a radius of 2.8 cm, and cover the plates to avoid contamination. Fill the six cylinders on each plate with dilutions of antibiotic containing the test levels (S1–S5 and U3) specified in the following paragraph. Incubate the plates as specified in Table 6 for 16–18 h, and remove the cylinders. Measure and record the diameter of each zone of growth inhibition to the nearest 0.1 mm.
Table 6
The standards (S1–S5) and a single test level of the sample (U3) corresponding to S3 of the standard curve, as defined in Standard solutions and Sample solutions will be used in the assay. For deriving the standard curve, fill alternate cylinders on each of three plates with the median test dilution (S3) of the standard and each of the remaining nine cylinders with one of the other four test dilutions of the standard. Repeat the process for the three test dilutions of the standard. For the sample, fill alternate cylinders on each of three plates with the median test dilution of the standard (S3), and fill the remaining nine cylinders with the corresponding test dilution (U3) of the sample.
Turbidimetric Method
Temperature control:
Use appropriately qualified and calibrated equipment to obtain the temperature ranges specified in Table 8. [Note—Temperature control can be achieved using either circulating air or water. The greater heat capacity of water lends it some advantage over circulating air. ]
Spectrophotometer:
Measuring absorbance or transmittance within a fairly narrow frequency band requires a suitable spectrophotometer in which the wavelength can be varied or restricted by use of 580nm or 530nm filters. Alternatively, a variablewavelength spectrophotometer can be used and set to a wavelength of 580 nm or 530 nm.
The instrument may be modified as follows:
Autozero the instrument with clear, uninoculated broth prepared as specified for the particular antibiotic, including the same amount of test dilution (including formaldehyde if specified) as found in each sample.
Either absorbance or transmittance can be measured while preparing inocula.
Apparatus:
Glass or plastic test tubes, e.g., 16 × 125 mm or 18 × 150 mm. [Note—Use tubes that are relatively uniform in length, diameter, and thickness and substantially free from surface blemishes and scratches. In the spectrophotometer, use matched tubes that are free from scratches or blemishes. Clean tubes thoroughly to remove all antibiotic residues and traces of cleaning solution. Sterilize tubes before use. ]
Standard solutions:
To prepare a stock solution, dissolve a quantity of the USP Reference Standard of a given antibiotic or the entire contents of a vial of USP Reference Standard, where appropriate, in the solvent specified in Table 7, and dilute to the required concentration. Store at 2–8, and use within the period indicated. On the day of the assay, prepare from the stock solution five or more test dilutions, the successive solutions increasing stepwise in concentration, usually in the ratio of 1:1.25. [Note—It may be necessary to use smaller ratios for the successive dilutions from the stock solution for the turbidimetric assay. ] Use the final diluent specified such that the median level of the standard (S3) has the concentration suggested in Table 7.
Sample solutions:
Assign an assumed potency per unit weight or volume to the unknown, and on the day of the assay prepare a stock solution in the same manner specified for the USP Reference Standard (Table 7). Dilute the sample stock solution in the specified final diluent at a nominal concentration equal to the median concentration of the standard (S3) as specified in Table 7.
Table 7
Inocula:
Suspend the test organism from a freshly grown slant or culture in 3 mL of sterile saline TS. Glass beads can be used to facilitate the suspension. Enterococcus hirae (ATCC 10541) and Staphylococcus aureus (ATCC 9144) are grown in a liquid medium, not on agar. Spread the saline suspension onto the surface of two or more agar plates (covering the entire surface) or onto the surface of a Roux bottle containing 250 mL of the specified medium (see Table 8). Incubate at the time and temperature specified in Table 8, or until growth is apparent.
After incubation, harvest the organism from the plates or Roux bottle with approximately 50 mL of sterile saline TS, using a sterile bent glass rod or sterile glass beads. Pipet the suspension into a sterile glass bottle. This is the harvest suspension.
Determine during method verification the quantity of harvest suspension that will be used as the inoculum, starting with the volume suggested in Table 8. Prepare also an extra S3 as a test of growth. Incubate the trial tests for the times indicated in Table 11. Adjust the quantity of inoculum daily, if necessary, to obtain the optimum concentration–response relationship from the amount of growth of the test organism in the assay tubes. At the completion of the specified incubation periods, tubes containing the median concentration of the standard should have absorbance values as specified in Table 9. Determine the exact duration of incubation by observing the growth in the reference concentration (median concentration) of the standard (S3).
Table 8
Table 9
Analysis:
On the day of the assay, prepare the necessary concentration of antibiotic by dilution of stock solutions of the standard and of each sample as specified under Standard solutions and Sample solutions. Prepare five test levels, each in triplicate, of the standard (S1–S5) and a single test level (U3), also in triplicate, of up to 20 samples corresponding to S3 (median concentration) of the standard.
Table 10
Place the tubes in test tube racks or other carriers. Include in each rack 1–2 control tubes containing 1 mL of the inoculum medium (see Table 8) but no antibiotic. Add the volumes of the standard and sample test dilutions as indicated in Table 10. Randomly distribute one complete set, including the controls, in a tube rack. Add the volume of inoculum specified in Table 10 to each tube in the rack in turn, and place the completed rack immediately in an incubator or a water bath maintained at the temperature specified in Table 8 and for the time specified in Table 11.
Table 11
After incubation, immediately inhibit the growth of the organism by adding 0.5 mL of dilute formaldehyde to each tube, except for tylosin. For tylosin, heat the rack in a water bath at 80–90 for 2–6 min or in a steam bath for 5–10 min, and bring to room temperature. Read absorbance or transmittance at 530 or 580 nm, analyzing one rack at a time.
Media and Solutions The media required for the preparation of test organism inocula are made from the ingredients listed herein. Minor modifications of the individual ingredients are acceptable; and reconstituted dehydrated media can be substituted, provided that the resulting media possess equal or better growthpromoting properties and give a similar standard curve response.
Media:
Dissolve the ingredients in water to make 1 L, and adjust the solutions with either 1 N sodium hydroxide or 1 N hydrochloric acid as required, so that after steam sterilization the pH is as specified.
Medium 1
Medium 2
Medium 3
Medium 4
Medium 5
Medium 8
Medium 9
Medium 10
Medium 11
Medium 13
Medium 19
Medium 32
Medium 34
Medium 35
Medium 36
Medium 39
Medium 40
Medium 41
Solutions
Buffers:
Prepare as directed in Table 12, or by other suitable means. The buffers are sterilized after preparation; the pH specified in each case is the pH after sterilization.
Table 12. Buffers
Other solutions:
See Reagents, Indicators, and Solutions.
Water:
Use Purified Water.
Saline:
Use saline TS.
Dilute formaldehyde:
Formaldehyde solution and water (1:3)
Calculations
Introduction:
Antibiotic potency is calculated by interpolation from a standard curve using a logtransformed straightline method with a leastsquares fitting procedure (see below for calculation details). The analyst must consider three essential concepts in interpreting antibiotic potency results:
The laboratory should predetermine in its standard operating procedures a maximum acceptable value for the confidence interval width. This maximum value should be determined during development and confirmed during validation or verification. If the calculated confidence interval width exceeds this limit, the analyst must perform additional independent potency determinations to meet the limit requirement. Note that the decision to perform additional determinations does not depend on the estimated potency but only on the uncertainty in that estimate as determined by the confidence interval width. Assay variability has a greater impact on the calculated confidence limit than does the number of independent potency determinations. As a result, the analyst should first consider decreasing variability to the extent possible before conducting potency determinations.
The following sections describe the calculations for determining antibiotic potency as well as for performing the confidence limit calculation. Methods for calculating standard error are also shown in order to allow estimates of assay variance. Where logarithms are used, any base log is acceptable. Appendix 1 provides formulas for hand calculations applicable when the concentrations are equally spaced in the log scale. Alternative statistical methods may be used if appropriately validated.
Cylinderplate assay:
This section details analysis of the sample data and determination of the potency of an unknown, using the cylinderplate assay.
Sample data:
Table 13 shows the data from one assay that will be used as an example throughout this section. For each of the 12 plates, zones 1, 3, and 5 are the reference concentration and the other three zones are for one of the other four concentrations, as shown. Other columns are needed for calculations and are explained below.
Step 1:
Perform initial calculations and variability suitability check.
For each set of three plates, average the nine reference values and average the nine standard values.
Example
(see Table 13)
15.867 = X (16.1, 15.6, …, 15.8)
14.167 = X(14.6, 14.1, …, 14.8)
For each set of three plates determine the standard deviation of the nine reference values and the standard deviation of the nine standard values. For each standard deviation, determine the corresponding relative standard deviation.
Example
(see Table 13)
0.200 = (16.1, …, 15.8)
1.3% = (0.200/15.867) × 100
0.324 = (14.6, …, 14.1)
2.3% = (0.324/14.167) × 100
For a variability suitability criterion, each laboratory should determine a maximum acceptable value for the relative standard deviation. If any of the eight relative standard deviations (four for the reference and four for the standard) exceed this predetermined maximum, the assay data are not suitable and should be discarded. [Note—The suggested limit for relative standard deviation is NMT 10%. ]
Step 2:
Perform a platetoplate variation correction.
This correction is applied to convert the average zone measurement obtained for each concentration to the value it would be if the average reference concentration measurement for that set of three replicate plates were the same as the value of the correction point:
XC = XS (XR P)
XC = corrected standard mean XS = original standard mean XR = reference mean P = correction point
Example:
For the first set of three plates in Table 13 (S1), the correction is:
14.022 = 14.167 (15.867 15.722) = 14.167 0.145
Step 3:
Determine the standard curve line.
Generate the standard curve line by plotting the corrected zone measurements versus the log of the standard concentration values. Calculate the equation of the standard curve line by performing a standard unweighted linear regression on these values, using appropriate software or the manual calculations of Appendix 1. [Note—Use either the natural log or the base 10 log to plot the standard curve and determine the regression equation; both provide the same final test result. ] Each laboratory should determine a minimum value of the coefficient of determination (%R2) for an acceptable regression. The regression is acceptable only if the obtained %R2 exceeds this predetermined value. [Note—The suggested limit for the percentage coefficient of determination is NLT 95%. ]
Table 13. Sample Data (CylinderPlate Assay)
Linear regression results
Standard curve line:
Z = [3.551 × ln(C)] + 9.978
Z = corrected zone measurement C = concentration
%R2 = 99.7
Sample potency determination:
To estimate the potency of the unknown sample, average the zone measurements of the standard and the zone measurements of the sample on the three plates used. Correct for platetoplate variation using the correction point determined above to obtain a corrected average for the unknown, U. [Note—An acceptable alternative to using the correction point is to correct using the value on the estimated regression line corresponding to the log concentration of S3. ] Use the corrected average zone measurement in the equation of the standard curve line to determine the log concentration of the sample, LU, by:
LU = (U a)/b
a = intercept of the regression line b = slope of the regression line To obtain the potency of the unknown, take the antilog of LU and multiply the result by any applicable dilution factor. This value can also be expressed as a percentage of the reference concentration value.
Example:
Corrected sample zone measurement (Table 13) = 15.522
Natural log of the sample concentration:
LU = (15.522 9.978)/3.551 = 1.561
Sample concentration:
CU = e1.561 = 4.765
Percentage of reference concentration:
Result = (4.765/5.000) × 100 = 95.3%
Turbidimetric assay:
This section details analysis of the sample data and determination of the potency of an unknown using the turbidimetric assay. The method assumes that the tubes are randomly distributed within the heat block or other temperature control device. If the device has a temperature profile that is not uniform, a randomized blocks design is preferred. In such a design, the rack is divided into areas (blocks) of relatively uniform temperature and at least one tube of each Standard concentration and of each unknown is placed in each area. The data analysis of a randomized block design is different from the following.
Sample data:
Table 15 shows the data from one assay that will be used for an example throughout this section. Other columns are needed for calculations and are explained below.
Table 15. Sample Data (Turbidimetric Assay)
Step 1:
Perform initial calculations and variability suitability check.
For each concentration (including the sample), average the three absorbance values.
Example:
See S1 in Table 15.
0.0125 = SD(0.8545, 0.8422, 0.8495)
The combined value is calculated by taking the square root of the average of the five variances:
0.0325 = {[(0.0062)2 + (0.0125)2 + (0.0640)2 + (0.0119)2 + (0.0272)2]/5}1/2
For a variability suitability criterion, each laboratory should determine a maximum acceptable combined standard deviation. If the combined standard deviation exceeds this predetermined maximum, the assay data are not suitable and should be discarded. [Note—The suggested limit for the combined standard deviation is NMT 10% of the average absorbance value across the five concentrations. ] If the number of replicates per concentration is at least five, then a relative standard deviation can be computed for each concentration after checking for outliers and compared to a maximum acceptable relative standard deviation. [Note—The suggested limit for the relative standard deviation is NMT 10%. ]
Step 2:
Determine the standard curve line.
Generate the standard curve line by plotting the average absorbance values versus the log of the standard concentration values. Calculate the equation of the standard curve line by performing an unweighted linear regression on these values using appropriate software or the manual calculations of Appendix 1. [Note—Use either the natural log or the base 10 log to plot the standard curve and determine the regression equation; both provide the same final test result. ] Each laboratory should determine a minimum value of the percentage coefficient of determination (%R2) for an acceptable regression. The regression is acceptable only if the %R2 value obtained exceeds this predetermined value. [Note—The suggested limit for the percentage coefficient of determination is NLT 90%. ]
Linear regression results
Standard curve line:
Absorbance = 2.2665 [0.7735 × log10(concentration)]
%R2 = 93.0%
Sample potency determination:
To estimate the potency of the unknown sample, average the three absorbance measurements to obtain an average for the unknown, U. Use this average measurement in the equation of the standard curve line to determine the log concentration of the unknown sample, LU, by:
LU = (U a)/b
a = intercept of the regression line b = slope of the regression line To obtain the potency of the unknown, take the antilog of LU and multiply the result by any applicable dilution factor. This value can also be expressed as a percentage of the reference concentration value.
Example:
Average sample absorbance (Table 15) = 0.7430.
log10(CU) = (0.7430 2.2665)/( 0.7735) = 1.9696
CU = 101.9696 = 93.2
Percentage of reference concentration = (93.2/100.0) × 100 = 93.2%
CU = concentration of the sample
Confidence limits and combination of assays calculations:
Because of interassay variability, three or more independent determinations are required for a reliable estimate of the sample potency. For each independent determination, start with separately prepared stock solutions and test dilutions of both the Standard and the sample, and repeat the assay of a given sample on a different day.
Given a set of at least three determinations of the unknown potency, use the method of Appendix 2 to check for any outlier values. This determination should be done in the log scale.
To obtain a combined estimate of the unknown potency, calculate the average, M, and the standard deviation of the accepted log potencies. [Note—Use either the natural log or the base 10 log. ] Determine the confidence interval for the potency as follows:
antilog[M t(0.05, N 1) × SD/N], antilog[M + t(0.05, N 1) × SD/N]
M = average SD = standard deviation N = number of assays t (0.05, N1) = the twosided 5% point of a Student's tdistribution with N1 degrees of freedom
Note—The t value is available in spreadsheets, statistics texts, and statistics software.
W = antilog{[t(0.05, N 1) × SD/N]}
W = halfwidth of the confidence interval Compare the halfwidth of the confidence interval to a predetermined maximum acceptable value. If the halfwidth is larger than the acceptance limit, continue with additional assays.
Example:
Suppose the sample is assayed four times, with potency results in the natural log scale of 1.561, 1.444, 1.517, and 1.535. Then:
N = 4
M = X(1.561, 1.444, 1.517, 1.535) = 1.514
SD = (1.561, 1.444, 1.517, 1.535) = 0.050
t = 3.182
The confidence interval in the log scale is
1.514 ± (3.182 × 0.050/4) = (1.434, 1.594)
Taking antilogs, the estimated potency is
e1.514 = 4.546
with a 95% confidence interval for the potency of e1.434, e 1.594 = (4.197, 4.924).
The confidence interval halfwidth to compare to an acceptance value is the ratio 4.924/4.546 = 1.083.
Appendix 1. Formulas for Manual Calculations of Regression and Sample Concentration If the concentrations are equally spaced in the logarithmic scale, the calculations can be performed using the following formula. Let:
Sk = mean corrected zone measurement (cylinderplate assay) or average absorbance value (turbidimetric assay) for standard set k k = 1, 2, 3, 4, 5 S = mean of the five Sk values Lk = logarithm of the kth concentration. [Note—Use either the natural log or the base 10 log. Slope of the regression line is calculated by: ]
b = (Yhigh Ylow)/(Xhigh Xlow)
Yhigh = 1/5(3S5 + 2S4 + S3 S1)
Ylow = 1/5(3S1 + 2S2 + S3 S5)
Xhigh = L5 Xlow = L1 Combine and simplify to:
b = (4S5 + 2S4 2S2 4S1)/[5(L5 L1)]
The log of the concentration of the sample is found using:
LU = Lreference + [(U – S)/b]
For example, using the data for the cylinderplate assay in Table 13 and natural logarithms:
b = [(4 × 17.222) + (2 × 16.511) (2 × 14.989) (4 × 14.020)]/{5[ln(7.81)] ln(3.2)} = 3.551
S = (14.020 + 14.989 + 15.722 + 16.511 + 17.222)/5 = 15.693
Natural log of sample concentration = ln(5) + [(15.522 15.693)/3.551] = 1.561
Sample concentration = e1.561 = 4.765
Appendix 2. Procedure for Checking for Outliers; Rejection of Outlying or Aberrant Measurements A measurement that is clearly questionable because of a failure in the assay procedure should be rejected, whether it is discovered during the measuring or tabulation procedure. The arbitrary rejection or retention of an apparently aberrant measurement can be a serious source of bias. In general, the rejection of measurements solely on the basis of their relative magnitudes is a procedure that should be used sparingly.
Each suspected potency measurement, or outlier, may be tested against the following criterion. This criterion is based on the variation within a single group of supposedly equivalent measurements from a normal distribution. On average, it will reject a valid observation once in 25 trials or once in 50 trials. Designate the measurements in order of magnitude from y1 to yN, where y1 is the candidate outlier, and N is the number of measurements in the group. Compute the relative gap by using Table A21, Test for Outlier Measurements, and the formulas below:
When N = 3 to 7:
G1 = (y2 y1)/(yN y1)
When N = 8 to 10:
G2 = (y2 y1)/(yN 1 y1)
When N = 11 to 13:
G3 = (y3 y1)/(yN 1 y1)
If G1, G2, or G3, as appropriate, exceeds the critical value in Table A21, Test for Outlier Measurements, for the observed N, there is a statistical basis for omitting the outlier measurement(s).
Table A21. Test for Outlier Measurements
Example:
Estimated potencies of sample in log scale = 1.561, 1.444, 1.517, 1.535.
Check lowest potency for outlier:
G1 = (1.517 1.444)/(1.561 1.444) = 0.624 < 0.889
Therefore 1.444 is not an outlier.
Check highest potency for outlier:
G1 = (1.561 1.535)/(1.561 1.444) = 0.222 < 0.889
Therefore 1.561 is not an outlier.
Outlier potencies should be marked as outlier values and excluded from the assay calculations. NMT one potency can be excluded as an outlier.USP35
Auxiliary Information—
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USP35–NF30 Page 74
Pharmacopeial Forum: Volume No. 36(5) Page 1239
