© 2015 APMIS. Published by John Wiley & Sons Ltd. DOI 10.1111/apm.12405
APMIS 123: 731–739
How to evaluate PCR assays for the detection of low-level DNA FREDERIK BANCH CLAUSEN,1 EMIL URHAMMER,2 KLAUS RIENECK,1 GRETHE RISUM KROG,1 LEIF KOFOED NIELSEN3 and MORTEN HANEFELD DZIEGIEL1,4 Department of Clinical Immunology, Copenhagen University Hospital, Copenhagen; 2Department of Development and Planning, Aalborg University, Copenhagen; 3Department of Technology, Faculty of Health and Technology, Metropolitan University College, Copenhagen; and 4Copenhagen University, Copenhagen, Denmark 1
Clausen FB, Urhammer E, Rieneck K, Krog GR, Nielsen LK, Dziegiel MH. How to evaluate PCR assays for the detection of low-level DNA. APMIS 2015; 123: 731–739. High sensitivity of PCR-based detection of very low copy number DNA targets is crucial. Much focus has been on design of PCR primers and optimization of the amplification conditions. Very important are also the criteria used for determining the outcome of a PCR assay, e.g. how many replicates are needed and how many of these should be positive or what amount of template should be used? We developed a mathematical model to obtain a simple tool for quick PCR assay evaluation before laboratory optimization and validation procedures. The model was based on the Poisson distribution and the Binomial distribution describing parameters for singleplex real-time PCR-based detection of lowlevel DNA. The model was tested against experimental data of diluted cell-free foetal DNA. Also, the model was compared with a simplified formula to enable easy predictions. The model predicted outcomes that were not significantly different from experimental data generated by testing of cell-free foetal DNA. Also, the simplified formula was applicable for fast and accurate assay evaluation. In conclusion, the model can be applied for evaluation of sensitivity of realtime PCR-based detection of low-level DNA, and may also assist in design of new assays before standard laboratory optimization and validation is initiated. Key words: qPCR; low-level DNA; cffDNA; Poisson; Mathematical model. Frederik Banch Clausen, Department of Clinical Immunology, Rigshospitalet, Copenhagen University Hospital, Blegdamsvej 9, DK-2100 Copenhagen, Denmark. e-mail: [email protected]
Quantitative real-time PCR (qPCR) is a powerful tool in molecular and diagnostic analysis (1, 2). In a variety of fields, including clinical immunology, microbiology, pathology, forensic science, or food quality assessment, detection of low-level DNA can be a challenge for a reliable DNA analysis (3, 4). Real-time PCR is also applied in noninvasive prenatal testing or diagnosis, where cell-free foetal DNA (cffDNA) is extracted from the blood from pregnant women and analysed for selected foetal gene targets, to predict, e.g. the foetal sex or the RhD blood type (5, 6). The cffDNA is present in small quantities (7), and this is a major challenge for the analytical sensitivity of qPCR-based assays which affects the diagnostic sensitivity and accuracy of any such qPCR-based assay. Therefore, there is a need for an efficient Received 13 March 2015. Accepted 19 April 2015
way to evaluate method setups for qPCR that are intended for detecting low-level DNA to facilitate robust methods for research and diagnostics. In particular, a simple model to estimate a theoretical limit of detection (LoD) before standard laboratory procedures of optimization and validation would be very valuable. In the guidelines for Minimum Information for Publication of Quantitative real-time PCR Experiments (MIQE) that aim to standardize analysis and reporting criteria for qPCR experiments, the LoD is defined as the lowest concentration at which 95% of the positive samples are detected (8). Importantly, it is stated that the most sensitive LoD is three DNA copies per PCR well (8). In an optimized and sensitive setup of qPCR, the detection limit is a single copy (9, 10). However, the central point regarding the analytical sensitivity is whether a single copy is present in the PCR well. 731
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The sampling of very few copies is expected to match a Poisson distribution. We have previously shown that low levels of DNA follow a Poisson distribution (11, 12). The Poisson distribution describes the probability that an event will take place when the frequency of that event occurring is very low (13). This is equivalent to the probability that a PCR well will contain a DNA copy based on the knowledge of the average number of DNA copies per sampling volume of a sample from a solution with a very low DNA concentration. The LoD of three copies per PCR derives from calculations based on the Poisson distribution where an average of three copies per PCR will generate a 95% chance that one PCR well will contain at least one DNA copy (9). This calculation of LoD concerns setups where all PCR reactions must be positive, and one alternative is to allow some negative PCR reactions of a set of total reactions. In noninvasive prenatal testing, the quantities of cffDNA have been reported as approximately ranging from 1 to 1000 copies per mL (14), depending on the detection method and the gestational age at blood sampling. Analysis of such quantities often leads to PCR analysis of extracted DNA in a concentration below an average of three copies per PCR well (12). Therefore, there is a specific need for a model that can describe LoDs lower than three copies per PCR, in such a way that the analytical sensitivity of a given assay can be evaluated or that a new setup can be designed according to a LoD necessary for a given sensitive assay. We present a novel mathematical model of lowlevel DNA and propose a new way to calculate the LoD, to evaluate assay sensitivity, or as a tool for exploring new assay design prior to laboratory optimization and validation procedures. We present the mathematical background of the model, we test its predictions compared with experimental data, and we provide examples of its applications. MATERIALS AND METHODS Samples In this study, we used anonymized, pooled, residual DNA material, from a routine DNA extraction from pregnant women’s plasma. Blood group testing on residual material from blood was approved by the Scientific-Ethical Committees for Copenhagen and Frederiksberg (KF 01283691), which waived the need for written consent.
based on the Poisson distribution (see below). A LoD can be assessed and evaluated as copies per PCR. From noninvasive prenatal testing, the model also adopts the term plasma equivalent per PCR (measured in mL) which calculates the fraction of maternal plasma that is analysed per PCR reaction (DNA template volume x plasma volume/ elution volume) (15). Thus, the model also describes the overall setup including DNA extraction and criteria for a positive result. The mathematics is based on assumptions that a single copy target (DNA copy) is necessary and sufficient for PCR amplification and will yield a positive PCR reaction, i.e. no false positive and no false negative PCR reactions occur, and that each DNA copy is distributed independently of the other copies. The latter assumption is reasonable for degraded or short-length DNA material, such as cffDNA. If a stochastic variable (X1) of DNA copies in a DNA sample follows the Poisson distribution, then the probability of pipetting no DNA copies into a PCR well is
PðX1 ¼ 0Þ ¼ ek
(1)
where k is the average number of DNA copies per PCR well. The probability of pipetting one or more DNA copies into a PCR well is then 1 – e-k. k depends on the DNA quantity in the original sample, the DNA extraction efficiency, and the plasma equivalent per PCR. Thus,
k ¼ ux
(2)
where φ is the plasma equivalent per PCR, and x is the number of DNA copies per mL extracted from the original sample. To calculate a theoretical LoD for a given setup, we must calculate the number of DNA copies per mL (x) that yields an average number of DNA copies per PCR well (k) that will result in a minimum of x positive PCR reactions of the total number of n PCR reactions. Where x of n (x/n) is the selected criteria for a positive test result. This situation can be described by the Binomial distribution. If we let the probability of a positive PCR be determined by the Poisson distribution, we can define a stochastic variable (X2) as the number of positive PCR reactions of the total number of PCR reactions, predicting the probability of an outcome of a minimum of x positive PCRs of a total of n PCRs with an estimated average of k copies per PCR well:
PðX2 xÞ ¼
n X n ð1 ek Þk ðek Þnk k
(3)
k¼x
Using [2], it follows that
Calculations of detection probability A mathematical prediction model was created to assess the sensitivity of a singleplex setup for detecting low-level DNA. On the basis of the average number of DNA copies per PCR well, a calculation of the LoD is possible. The model includes a calculation of the detection probability
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PðX2 xÞ ¼
n X n ð1 eux Þk ðeux Þnk k
(4)
k¼x
Probabilities of positive detection can be calculated for increasing k, and curves can be generated of accumulating © 2015 APMIS. Published by John Wiley & Sons Ltd
RELIABLE DETECTION OF LOW-LEVEL DNA
detection probabilities, thus, ultimately generating a measurement of the probability of detecting the DNA in a positive sample with low-level DNA. We defined the 95% LoD (LoD0.95) as the number of DNA copies at which a 95% probability of detection is reached (expressed either as x copies per mL or k copies per PCR), with a given set of criteria for positive (x/n). This uniform model is applicable despite differences in method setup. The model can be set up in Microsoft Excel to calculate P, thus, estimating the chance of a positive result under different conditions and at different levels of DNA. See Table S1, for an Excel setup. We used MATLABÒ (version 7.14, The MathWorks Inc., Natick, Massachusetts, USA) to isolate x, given any value of P, x, n, and φ. We designed a script in MATLABÒ based on an F-solver. In addition, we aimed to find an approximation formula as a tool for a quick and easy evaluation of a given setup. For these investigations, we used curve fitting on groupings of k values related to (n/x) or n using Microsoft Excel 2007. The approximation formulas were tested against model data (using [3]) with Spearman Correlation, and Wilcoxon’s matched pairs signed rank test where a p value less than 0.05 was considered significant. Statistical calculations were performed using GraphPad Prism 5.02 (GraphPad Software, Inc., San Diego, CA).
DNA analysis To test the model-based in silico predictions of LoDs, we analysed repeated detection of low-level DNA in an average concentration of 0.5–3 DNA copies per PCR reaction and compared the outcome to model-based predictions for selected setups of x/n. We used the detection of diluted cffDNA as our test system. CffDNA was extracted from plasma from RhD negative pregnant women in gestational week 25 using the automated QIAsymphony SP extraction system (Qiagen, Basel, Switzerland) using the QIAsymphony DSP Virus/Pathogen Midi Kit using carrier RNA, and eluted in AE buffer. The PCR assay targeted the RhD gene, RHD (accession number NT_004610.19). Extracted RHD positive cffDNA was pooled and stored at 20 °C until analysis. The pool was diluted 1:10, and the cffDNA concentration was estimated using a five-point standard curve comprising reactions of 10 ng, 2 ng, 400 pg, 80 pg, and 16 pg DNA per PCR from an RHD hemizygous individual, detected by an RHD exon 7 assay (11). Then new dilutions of the pool were made with DNA concentrations equivalent to an average of 0.5, 1, 2, or 3 RHD positive DNA copies per PCR well. Dilutions were re-quantified by counting negative PCR reactions (N0) of a total (NT) of 96 reactions. The average number of copies per PCR reaction was calculated via k = ln(N0/NT) (13). Theoretical data were then calculated by the model based on this estimated k. A new real-time PCR analysis of the same dilution of DNA in 96 reactions was then carried out to obtain empirical data on the estimated k. Comparison of theoretical and empirical data was performed using Wilcoxon’s matched pairs signed rank test where a p value less than 0.05 was considered significant. The QIAgility instrument (Qiagen Inc.) was used for automatic PCR dispensing into MicroAmpÒ Optical 96-Well Reaction Plates (Applied BioSystems,
© 2015 APMIS. Published by John Wiley & Sons Ltd
Foster City, CA, USA). Real-time PCR was conducted using an ABI 7500 detection system (Applied BioSystems) with TaqMan chemistry. PCR for detection of RHD exon 7 was set up with 10 lL template volume in a total volume of 25 lL with Universal Master Mix. Final primer concentration was 900 nM and final concentration of the FAM- labelled probe was 100 nM. Primers and probe were synthesized by Eurofins MWG Operon (Edersberg, Germany). PCR profile was as previously described (11). Results were evaluated with the 7500 System SDS software version 1.4 from Applied BioSystems using auto baseline and a fixed threshold of 0.2. A PCR was considered positive if the cycle threshold (CT) value was below 42. Standard curve characteristics for the exon 7 assay were: y = 3.397x + 42.13, R2 = 0.998; PCR efficiency was 0.97.
RESULTS Detection probability
We developed a mathematical prediction model based on a combination of the Poisson distribution and the Binomial distribution. With this model, we were able to calculate LoDs for a range of scenarios to investigate the significance of the plasma equivalent per PCR and different applications of replicate PCR reactions and criteria for positive results in assays intended for detection of low-level DNA. We showed that increasing the plasma equivalent per PCR by a given factor improves the LoD0.95 by the same factor, see Fig. 1. In Fig. 2, the plasma equivalent per PCR is fixed at 0.1 mL showing different setup examples with different numbers of replicates and criteria for a positive result. It shows that if the criteria for a positive result allow at least one reaction to be negative (x < n), this will generate a lower LoD0.95 compared to a setup where all reactions must be positive (x = n), see Fig. 2. The PCR setups with different criteria for a positive result (x/n) were ranked by their sensitivity, calculated as the minimum average number of DNA copies per PCR resulting in a 95% probability of detection of a positive sample (k0.95), shown in Fig. 3. In addition, we found that an approximation calculation could serve as a quick calculation to guide choices of setup, see Fig. 4. For x < n and n < 8, the approximation of k0.95 was k0:95 ¼ 3:12ðn=xÞ1:15
(5)
with R2 = 0.969. From this approximation, we simplified the formula even further: k0:95 ¼ 3x=n
(6)
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Fig. 1. Predicting limits of detection (LoD) for three different plasma equivalents per PCR (50, 100 fixed criterion of at least 2/3 positive PCR reactions for a positive result. The LoD was defined as copies per mL resulting in a predicted 95%-probability of detection. The LoD was 40 DNA copies plasma equivalent per PCR of 50 lL. The LoD was 20 DNA copies for 100 lL, and 10 DNA copies
and 200 lL), using a the number of DNA per mL plasma for a for 200 lL.
Fig. 2. Predicting limits of detection (LoD) for different setup concerning replicate PCR reactions and criteria for positive result. The plasma equivalent per PCR was fixed at 100 lL. The LoD was defined as the number of DNA copies per mL resulting in a predicted 95%-probability of detection for each setup. Using 3/3 positive PCR reactions for a positive result yielded a LoD of 41 DNA copies per mL plasma (with a plasma equivalent per PCR of 100 lL). The LoD was 37 DNA copies for a criterion of 2/2, and the LoD was 20 DNA copies for at least 2/3, and 14 DNA copies for at least 2/4.
Using this simplified k, an approximated LoD0.95 in copies per mL (x0.95) could be calculated via [2] as: x0:95 ¼ 3x=nu
(7)
For x = n, k0:95 ¼ 0:99lnðxÞ þ 2:99 with R2 = 1. 734
(8)
A simplified approximation yielded an LoD0.95 in copies per mL as: x0:95 ¼ ðlnðxÞ þ 3Þ=u
(9)
Values of x0.95 obtained with different setups of x/n were calculated with all methods and tested for significant difference. For x = n, data from the simplified approximation were significantly different from model data generated in Microsoft Excel (Wilcoxon’s matched pairs signed rank test, p < 0.0001; © 2015 APMIS. Published by John Wiley & Sons Ltd
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Fig. 3. Different real-time PCR setups ranked according to the minimum average number of DNA copies per PCR at which a 95% probability of detection was reached (k0.95). Six is the maximum of total reactions presented.
Empirical test of model
Fig. 4. Curve fitting was applied to identify a simple relationship between x/n and k. Approximation: Fitted curve with corresponding curve statistics. Simplified approximation: The simplified approximation is shown in bold grey. Note that y = 3x1 is equal to k = 3x/n.
Spearman rs = 1.0000; n = 25) due to a systematic higher estimate of DNA copies, within 0.5 copies. With x < n at n < 8 there was no significant difference between model data generated from MATLABÒ and Microsoft Excel (Wilcoxon’s matched pairs signed rank test, p = 0.68; Spearman rs = 1.0000; n = 25) and no significant difference between model data generated from MATLABÒ and from the simplified approximation for x < n (Wilcoxon’s matched pairs signed rank test, p = 0.11; Spearman rs = 0.9935; n = 22). © 2015 APMIS. Published by John Wiley & Sons Ltd
Dilutions of low-level DNA were tested with realtime PCR, and the results were compared to model data generated in Microsoft Excel. With this experiment, we investigated whether empirical data obtained from laboratory testing would be in proximity of the data predicted by the model. Thus, given an estimate of k, we calculated a theoretical percentage (and a 95% confidence interval) of possible sets of replicate PCRs that would be positive and compared this with the observed percentage. We looked at assay setups with criteria for positive (x/n) of x/2 or x/3. All observed data fell within the 95% confidence interval, see Fig. 5. Overall, there was no statistically significant difference between the observed and the predicted frequencies of positive results (Wilcoxon’s matched pairs signed rank test, p = 0.8438; Spearman rs = 1.0000; n = 6). DISCUSSION We present a Poisson-based mathematical model for theoretical evaluation of singleplex setup choices for real-time PCR intended for reliable detection of low-level DNA, prior to standard laboratory optimization and validation procedures. Parameters that can be evaluated are the number of replicate PCR reactions, criteria for a positive result, and the plasma equivalent per PCR. In addition, we present a quick and easy approximation formula for rapid
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A
B
Fig. 5. Model predictions of outcomes of positive results with different criteria for positive (x/n) compared with empirical data obtained through PCR analysis, shown in percent. (A) Setups of 1/2 and 2/2 were analysed at k = 0.8, k = 1.8, and k = 3 DNA copies per PCR. (B) Setups of 1/3, 2/3, and 3/3 were analysed at k = 0.7, k = 1.8, and k = 3.5 DNA copies per PCR. Model predictions are presented with error bars representing 95% confidence intervals. Empirical PCR data are designated as observed. One experiment was carried out for each level of k, thus, representing a total of six experiments. Each experiment consisted of two PCR run, one for quantification of k, and, subsequently, one for analysis of predicted outcomes of x/n.
evaluation of setups for x < n (as well as a formula for x = n). For a new assay design, the consequence of different choices can be explored before initiating experimental work. Overall, these choices are central to the analytical sensitivity, and hence the diagnostic sensitivity and robustness of any given realtime PCR assay, in addition to other, well-known factors influencing the PCR assay. We based the model on the Poisson distribution of low-level DNA in a sample and the Binomial distribution of the PCR outcome as either positive or negative. In combination, this enabled us to describe the probability of obtaining a defined minimum number of positive PCRs based on the number of total PCR wells and the knowledge of the average number of DNA copies available in the sample. Importantly, the model allowed for the calculation of LoDs below an average of three copies per PCR. We presented calculations and experiments to demonstrate the applicability of the model. Based solely on mathematical work, it should be noted that the model does not aim to take into account certain physicochemical variables such as type of polymerase, inhibition from substances from the DNA extraction, or variations due to extraction technique. The model simply assesses the consequences of selected setup choices. In addition, a central assumption is that every present DNA copy is amplified, i.e. that we are in the range of singlecopy detection. This assumption is often fulfilled in practice. The technique is capable of detecting a single DNA copy (7, 16, 17). And as the number of positive reactions follows the Poisson distribution (10–12,18), and each copy is counted in the Poisson distribution, this implies directly that each copy will be amplified in the real detection system. 736
To investigate the applicability of our mathematical model, we tested the model-based predictions against empirical data obtained at 0.7–3.5 DNA copies per PCR. Exemplified by a selected number of setups, we demonstrated that the model predicts outcomes that are equal to those observed through experimental detection of low-level DNA. It shows that the model is solid enough for predicting realistic probabilities for outcomes of x/n given a certain known average level of DNA per PCR (k). With a k below 5, the distribution will match a Poisson distribution (13). Therefore, we could apply the k = –ln(N0/NT) formula, derived from Poisson, for quantification. Consequently, these experiments represented the testing of our model’s predictions in relation to experimental outcomes only under Poisson conditions. The model can be set up in Microsoft Excel or in, e.g. MATLAB. As an alternative, we developed a very simplified approximation formula of x0.95 = 3x/nφ (for x < n) that provides a quick assessment of any of the included parameters. The simple approximation formula was shown to fit the models from Microsoft Excel and MATLAB when n was kept below 8. Thus, the formula is simplified and limited; yet the applicability should be highly useful. In most cases, assay sensitivity can be calculated by head. See Box 1 for examples of how to use the formula. Using the formula to guide an assay setup should be combined with other considerations. For example, it would not be a good idea to use only one positive PCR reaction of many replicates as the criterion for a positive test outcome, as false positive reactions may occur, and such reactions would complicate the interpretation of the results. Overall, assessing the mathematical sensitivity via this formula may prevent wasting © 2015 APMIS. Published by John Wiley & Sons Ltd
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Box 1. Examples of how to use the simplified approximation formula
x0:95 ¼ 3x=nu We developed the approximation formula as a tool to enable a quick evaluation and design of PCR setups that are intended for reliable detection of low-level DNA. Abbreviations x0.95: estimated limit of detection (LoD) of DNA copies per mL sample with positive test outcomes in 95% of the cases; x: the number of positive PCR reactions; n = the number of total PCR reactions; x/n = the criterion for a positive test outcome; φ = the plasma-equivalent per PCR (mL), (plasma vol 9 template vol/elution vol). Examples of how to use the formula: Design a setup to detect one DNA copy per mL Aiming to detect only one DNA copy is an extreme example but demonstrates the use of the formula. Different setups can be used. If DNA is extracted from 5 mL, using an elution volume of 50 lL, and a template volume of 15 lL, then the plasma-equivalent per PCR, φ = (5 9 0.015/0.05) = 1.5. Then 1 = (3x)/(n 9 1.5) ⇒ (1.5/3) = (x/n) ⇒ 1/2 = x/n. Thus, the PCR setup could be two PCR reactions, with one positive PCR reaction as the criterion for a positive test outcome. Alternatively, if the DNA extraction is carried out from only 1 mL, using 50 lL elution, and a template volume of 10 lL, then φ = (1 9 0.01/0.05) = 0.2. Then 1 = (3x)/(n 9 0.2) ⇒ (0.2/3) = (x/n) ⇒ 1/6 = x/n. (This may, however, not be a wise setup due to the risk of false positive reactions interferring with the interpretation of the result). Find the LoD of a given assay setup Already known about the assay setup: DNA extraction volume = 1 mL, elution volume = 50 lL, template volume = 5 lL, x = 2, and n = 4. Calculations: The plasma-equivalent per PCR, φ = 1 9 0.005/0.05 = 0.1 mL. The LoD for the assay setup, x0.95 = (3 9 2)/(4 9 0.1) = 60/4 = 15 copies per mL. (This is an approximation of the exact LoD of 13.9 copies per mL). Design a setup when a parameter is fixed For practical or economical reasons, a setup may be limited by different requirements causing one or more parameters to be fixed at certain values. In general, the nonfixed values can be adjusted to improve the LoD, see below. Choose the number of PCR replicates If a LoD of 10 copies per mL is required and the plasma-equivalent is fixed at 0.2 mL, how many PCR replicates should be chosen to reach the required LoD? The following can be calculated: 10 = (3x)/(n 9 0.2) ⇒ 2 = 3x/n ⇔ 2/3 = x/n. Thus, one setup could be x = 2 and n = 3 as the criterion for a positive test outcome, i.e. at least 2 positive of 3 PCR replicates (the exact LoD is 10 copies per mL). Alternatively, one could choose 4 positive of 6 PCR reactions (the exact LoD is 9.38 copies per mL). Find the best plasma-equivalent per PCR If the PCR setup is fixed with 3 PCR reactions, and the criterion for a positive test outcome is that 2 PCR reactions must be positive, one can adjust the plasma-equivalent per PCR. Here, we can make different choices to secure a suitable LoD. We can calculate the following: x0.95 = (3 9 2)/(3 9 φ) ⇒ x0.95 = 2/φ. Thus, a plasma-equivalent per PCR of 0.2 would give a LoD of 10 copies per mL. If practically possible, we can choose an extraction volume of 2 mL, an elution volume of 50 lL, and a template volume of 5 lL, giving φ = 0.2. Other combinations of extraction, elution, and template volumes can also be used.
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Box 1 Continued
If x = n If the criterion for a positive test outcome is x = n, the LoD can be calculated using the formula: x0.95 = (ln(x) + 3)/φ. If, for example, the criterion is 2 positive of 2 PCR reactions and the plasma-equivalent per PCR is 0.15 mL, then the LoD is: x0.95 = (ln(2) + 3)/φ ⇒ x0.95 = (3.693/0.15) = 24.6 copies per mL.
valuable time in the laboratory. Assay adjustments can be performed to obtain a required LoD, before proceeding with standard laboratory work. For x = n, the statistically significant difference observed between the simplified model and the model in Microsoft Excel was due to a systematic higher DNA copy estimate, but only within 0.5 copies. This imprecision is of no practical consequence and acceptable for a simplified model. For example, if x = n = 3, then k0.95 = ln(3)+3 = 4.1 according to our simple model, corresponding to a LoD0.95 of 4.19 which was reported for the same setup (16). The ranking of k0.95 in Fig. 3 demonstrated the limit of three copies per PCR as the LoD for a setup where all replicates are demanded positive (x = n). This is in line with the observations from previously described theoretical limits for qPCR (8, 9). With our simple model for x = n, it is evident that k = 3 for n = 1, as ln(1) = 0. As seen for x = n in Fig. 3, k will increase as n increases. We presented our model for singleplex assay setups. It will be of great value to develop a similar model for multiplex setups. The possibility of evaluating the reliability of a given PCR setup is very important and may result in improved experimental design and increased speed to obtain a robust and accurate PCR-based assay. For testing of cffDNA, this is very important because the low quantities of cffDNA may hamper DNA detection especially in first-trimester analysis (6, 19). In addition, a high degree of variation and fluctuation (20) may also cause low levels later in pregnancy, so it is central that an assay setup is thoroughly evaluated and optimized, in the first as well as the second trimester of pregnancy. For everyone engaged in PCR analysis of low-level DNA, especially in a clinical setting, we recommend using our model to evaluate the mathematically determined assay sensitivity by estimating the LoD, or to include the model as a sensitivity assessment tool during design of new real-time PCR assays prior to extensive laboratory analysis. We recommend using the model as a tool to revise a PCR setup in general to improve or simplify the assay, or to revise the setup from an economical point of view. In our laboratory, the formula was 738
crucial in developing the routine antenatal assay to detect foetal RHD in maternal plasma, currently used in the Copenhagen region of Denmark (21–23). The track record of the assay demonstrates that the assay is extremely robust and highly sensitive, with only one false negative result in about 10,000 samples tested. In addition to the mathematical issues dealt with in this study, several other factors are central to the assay sensitivity of real-time PCR-based analysis of low-level DNA. For example, reliable analysis of cffDNA is affected by pre-analytical handling and sample processing (23–27), PCR-amplicon size (28, 29), DNA target repetition (12, 30, 31), and perhaps even nucleosome positioning (32). In conclusion, we have presented data and a mathematical description with formulas that enable easy evaluation of several important aspects of realtime PCR-based assays. The mathematical tool can help scientists to decide a specific setup for detection of low-level DNA by quickly evaluating a number of considerations that need to be taken into account before developing a new PCR based assay and prior to extensive laboratory optimization and validation procedures. Our simple mathematical model may assist laboratories in improving their assay sensitivity, facilitating a robust detection. CONFLICT OF INTEREST None declared. The authors acknowledge the contribution by Lone Vandsø Jørgensen and the staff from The Laboratory of Blood Genetics, Department of Clinical Immunology, Copenhagen University Hospital.
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3. Kay S, Van den Eede G. The limits of GMO detection. Nature 2001;19:405. 4. Pansiot J, Chaouachi M, Cavellini L, Romaniuk M, Ayadi M, Bertheau Y, et al. Development of two screening duplex PCR assays for genetically modified organism quantification using multiplex real-time PCR master mixes. Eur Food Res Technol 2011;232:327–34. 5. Hill M, Finning K, Martin P, Hogg J, Meaney C, Norbury G, et al. Non-invasive prenatal determination of fetal sex: translating research into clinical practice. Clin Genet 2011;80:68–75. 6. Clausen FB, Damkjær MB, Dziegiel MH. Noninvasive fetal RhD genotyping. Transfus Apher Sci 2014;50:154–62. 7. Lo YMD, Tein MSC, Lau TK, Haines CJ, Leung TN, Poon PM, et al. Quantitative analysis of fetal DNA in maternal plasma and serum: implications for noninvasive prenatal diagnosis. Am J Hum Genet 1998;62:768–75. 8. Bustin SA, Benes V, Garson JA, Hellemans J, Huggett J, Kubista M, et al. The MIQE Guidelines: Minimum information for publication of quantitative realtime PCR experiments. Clin Chem 2009;55:611–22. 9. Wittwer CT, Kusukawa N. Real-Time PCR. In: Persing H, Tenover FC, Versalovic J, Tang YW, Unger ER, Relman DA, White TJ, editors. Molecular Microbiology: Diagnostic Principles and Practice. Washington, DC, USA: ASM Press, 2004: 71–84. 10. Lagally ET, Medintz I, Mathies RA. Single-molecule DNA amplification and analysis in an integrated microfluidic device. Anal Chem 2001;73:565–70. 11. Clausen FB, Krog GR, Rieneck K, Nielsen LK, Lundquist R, Finning K, et al. Reliable test for prenatal prediction of fetal RhD type using maternal plasma from RhD negative women. Prenat Diagn 2005;25:1040–4. 12. Clausen FB, Krog GR, Rieneck K, R asmark EEF, Dziegiel MH. Evaluation of two real-time multiplex PCR screening assays detecting fetal RHD in plasma from RhD negative women to ascertain the requirement for antenatal RhD prophylaxis. Fetal Diagn Ther 2011;29:155–63. 13. Wang Z, Spadoro J. Determination of target copy number of quantitative standards used in PCR-based diagnostic assays. In: Ferre F, editor. Gene Quantification. Boston: Birkh€ auser, 1998: 31–41. 14. Finning KM, Martin PG, Soothill PW, Avent ND. Prediction of fetal D status from maternal plasma: introduction of a new noninvasive fetal RHD genotyping service. Transfusion 2002;42:1079–85. 15. Legler TJ, M€ uller SP, Haverkamp A, Grill S, Hahn S. Prenatal RhD testing: a review of studies published from 2006 to 2008. Transfus Med Hemother 2009;36:189–98. 16. Zimmermann B, El-Sheikhah A, Nicolaides K, Holzgreve W, Hahn S. Optimized real-time quantitative PCR measurement of male fetal DNA in maternal plasma. Clin Chem 2005;51:1598–604. 17. Zhang J, Fidler C, Murphy MF, Chamberlain PF, Sargent IL, Redman CW, et al. Determination of fetal RhD status by maternal plasma DNA analysis. Ann N Y Acad Sci 2000;906:153–5. 18. Rossmanith P, Wagner M. A novel poisson distribution-based approach for testing boundaries of realtime PCR assays for food pathogen quantification. J Food Prot 2011;74:1404–12. © 2015 APMIS. Published by John Wiley & Sons Ltd
19. Akolekar R, Finning K, Kuppusamy R, Daniels G, Nicolaides KH. Fetal RHD genotyping in maternal plasma at 11–13 weeks of gestation. Fetal Diagn Ther 2011;29:301–6. 20. Hahn S, Zhong XY, B€ urk MR, Troeger C, Kang A, Holzgreve W. Both maternal and fetal cell-free DNA in plasma fluctuate. Ann NY Acad Sci 2001;945:141–4. 21. Clausen FB, Christiansen M, Steffensen R, Jørgensen S, Nielsen C, Jakobsen MA, et al. Report of the first nationally implemented clinical routine screening for fetal RHD in D- pregnant women to ascertain the requirement for antenatal RhD prophylaxis. Transfusion 2012;52:752–8. 22. Clausen FB, Steffensen R, Christiansen M, Rudby M, Jakobsen MA, Jakobsen TR, et al. Routine noninvasive prenatal screening for fetal RHD in plasma of RhD-negative pregnant women-2 years of screening experience from Denmark. Prenat Diagn 2014;34:1–6. 23. Clausen FB, Jakobsen TR, Rieneck K, Krog GR, Nielsen LK, Tabor A, et al. Pre-analytical conditions in non-invasive prenatal testing of cell-free fetal RHD. PLoS One 2013;8(10):1–8. 24. Lui YYN, Lo YMD. Circulating DNA in plasma and serum: biology, preanalytical issues and diagnostic applications. Clin Chem Lab Med 2001;40:962–8. 25. M€ uller SP, Bartels I, Stein W, Emons G, Gutensohn K, Legler TJ. Cell-free fetal DNA in specimen from pregnant women is stable up to 5 days. Prenat Diagn 2011;31:1300–4. 26. Barrett AN, Zimmermann BG, Wang D, Holloway A, Chitty LS. Implementing prenatal diagnosis based on cell-free fetal DNA: accurate identification of factors affecting fetal DNA yield. PLoS One 2011;6(10):1–8. 27. Barrett AN, Thadani HA, Laureano-Asibal C, Ponnusamy S, Choolani M. Stability of cell-free DNA from maternal plasma isolated following a single centrifugation step. Prenat Diagn 2014;34:1283–88. 28. Grootkerk-Tax MGHM, Soussan AA, de Haas M, vanMaaskant- WIJK PA, dervan SCHOOT CE. Evaluation of prenatal RHD typing strategies on cell-free fetal DNA from maternal plasma. Transfusion 2006;46:2142–8. 29. Sikora A, Zimmermann BG, Rusterholz C, Birri D, Kolla V, Lapaire O, et al. Detection of increased amounts of cell-free fetal DNA with short PCR amplicons. Clin Chem 2010;56:136–8. 30. Hansen MH, Clausen FB, Dziegiel MH. Increased Ychromosome detection by SRY duplexing. Fetal Diagn Ther 2012;31:185–90. 31. Jin S, Lin XM, Law H, Kwek KY, Yeo GS, Ding C. Further improvement in quantifying male fetal DNA in maternal plasma. Clin Chem 2012;58:465–8. 32. Kaplan N, Moore IK, Fondufe-Mittendorf Y, Gossett AJ, Tillo D, Field Y, et al. The DNA-encoded nucleosome organization of a eukaryotic genome. Nature 2009;458:362–6.
SUPPORTING INFORMATION Additional Supporting Information may be found in the online version of this article: Table S1. Calculate the likelihood (P) of a positive test outcome for a given set of parameters.
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How to evaluate PCR assays for the detection of low-level DNA FREDERIK BANCH CLAUSEN,1 EMIL URHAMMER,2 KLAUS RIENECK,1 GRETHE RISUM KROG,1 LEIF KOFOED NIELSEN3 and MORTEN HANEFELD DZIEGIEL1,4 Department of Clinical Immunology, Copenhagen University Hospital, Copenhagen; 2Department of Development and Planning, Aalborg University, Copenhagen; 3Department of Technology, Faculty of Health and Technology, Metropolitan University College, Copenhagen; and 4Copenhagen University, Copenhagen, Denmark 1
Clausen FB, Urhammer E, Rieneck K, Krog GR, Nielsen LK, Dziegiel MH. How to evaluate PCR assays for the detection of low-level DNA. APMIS 2015; 123: 731–739. High sensitivity of PCR-based detection of very low copy number DNA targets is crucial. Much focus has been on design of PCR primers and optimization of the amplification conditions. Very important are also the criteria used for determining the outcome of a PCR assay, e.g. how many replicates are needed and how many of these should be positive or what amount of template should be used? We developed a mathematical model to obtain a simple tool for quick PCR assay evaluation before laboratory optimization and validation procedures. The model was based on the Poisson distribution and the Binomial distribution describing parameters for singleplex real-time PCR-based detection of lowlevel DNA. The model was tested against experimental data of diluted cell-free foetal DNA. Also, the model was compared with a simplified formula to enable easy predictions. The model predicted outcomes that were not significantly different from experimental data generated by testing of cell-free foetal DNA. Also, the simplified formula was applicable for fast and accurate assay evaluation. In conclusion, the model can be applied for evaluation of sensitivity of realtime PCR-based detection of low-level DNA, and may also assist in design of new assays before standard laboratory optimization and validation is initiated. Key words: qPCR; low-level DNA; cffDNA; Poisson; Mathematical model. Frederik Banch Clausen, Department of Clinical Immunology, Rigshospitalet, Copenhagen University Hospital, Blegdamsvej 9, DK-2100 Copenhagen, Denmark. e-mail: [email protected]
Quantitative real-time PCR (qPCR) is a powerful tool in molecular and diagnostic analysis (1, 2). In a variety of fields, including clinical immunology, microbiology, pathology, forensic science, or food quality assessment, detection of low-level DNA can be a challenge for a reliable DNA analysis (3, 4). Real-time PCR is also applied in noninvasive prenatal testing or diagnosis, where cell-free foetal DNA (cffDNA) is extracted from the blood from pregnant women and analysed for selected foetal gene targets, to predict, e.g. the foetal sex or the RhD blood type (5, 6). The cffDNA is present in small quantities (7), and this is a major challenge for the analytical sensitivity of qPCR-based assays which affects the diagnostic sensitivity and accuracy of any such qPCR-based assay. Therefore, there is a need for an efficient Received 13 March 2015. Accepted 19 April 2015
way to evaluate method setups for qPCR that are intended for detecting low-level DNA to facilitate robust methods for research and diagnostics. In particular, a simple model to estimate a theoretical limit of detection (LoD) before standard laboratory procedures of optimization and validation would be very valuable. In the guidelines for Minimum Information for Publication of Quantitative real-time PCR Experiments (MIQE) that aim to standardize analysis and reporting criteria for qPCR experiments, the LoD is defined as the lowest concentration at which 95% of the positive samples are detected (8). Importantly, it is stated that the most sensitive LoD is three DNA copies per PCR well (8). In an optimized and sensitive setup of qPCR, the detection limit is a single copy (9, 10). However, the central point regarding the analytical sensitivity is whether a single copy is present in the PCR well. 731
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The sampling of very few copies is expected to match a Poisson distribution. We have previously shown that low levels of DNA follow a Poisson distribution (11, 12). The Poisson distribution describes the probability that an event will take place when the frequency of that event occurring is very low (13). This is equivalent to the probability that a PCR well will contain a DNA copy based on the knowledge of the average number of DNA copies per sampling volume of a sample from a solution with a very low DNA concentration. The LoD of three copies per PCR derives from calculations based on the Poisson distribution where an average of three copies per PCR will generate a 95% chance that one PCR well will contain at least one DNA copy (9). This calculation of LoD concerns setups where all PCR reactions must be positive, and one alternative is to allow some negative PCR reactions of a set of total reactions. In noninvasive prenatal testing, the quantities of cffDNA have been reported as approximately ranging from 1 to 1000 copies per mL (14), depending on the detection method and the gestational age at blood sampling. Analysis of such quantities often leads to PCR analysis of extracted DNA in a concentration below an average of three copies per PCR well (12). Therefore, there is a specific need for a model that can describe LoDs lower than three copies per PCR, in such a way that the analytical sensitivity of a given assay can be evaluated or that a new setup can be designed according to a LoD necessary for a given sensitive assay. We present a novel mathematical model of lowlevel DNA and propose a new way to calculate the LoD, to evaluate assay sensitivity, or as a tool for exploring new assay design prior to laboratory optimization and validation procedures. We present the mathematical background of the model, we test its predictions compared with experimental data, and we provide examples of its applications. MATERIALS AND METHODS Samples In this study, we used anonymized, pooled, residual DNA material, from a routine DNA extraction from pregnant women’s plasma. Blood group testing on residual material from blood was approved by the Scientific-Ethical Committees for Copenhagen and Frederiksberg (KF 01283691), which waived the need for written consent.
based on the Poisson distribution (see below). A LoD can be assessed and evaluated as copies per PCR. From noninvasive prenatal testing, the model also adopts the term plasma equivalent per PCR (measured in mL) which calculates the fraction of maternal plasma that is analysed per PCR reaction (DNA template volume x plasma volume/ elution volume) (15). Thus, the model also describes the overall setup including DNA extraction and criteria for a positive result. The mathematics is based on assumptions that a single copy target (DNA copy) is necessary and sufficient for PCR amplification and will yield a positive PCR reaction, i.e. no false positive and no false negative PCR reactions occur, and that each DNA copy is distributed independently of the other copies. The latter assumption is reasonable for degraded or short-length DNA material, such as cffDNA. If a stochastic variable (X1) of DNA copies in a DNA sample follows the Poisson distribution, then the probability of pipetting no DNA copies into a PCR well is
PðX1 ¼ 0Þ ¼ ek
(1)
where k is the average number of DNA copies per PCR well. The probability of pipetting one or more DNA copies into a PCR well is then 1 – e-k. k depends on the DNA quantity in the original sample, the DNA extraction efficiency, and the plasma equivalent per PCR. Thus,
k ¼ ux
(2)
where φ is the plasma equivalent per PCR, and x is the number of DNA copies per mL extracted from the original sample. To calculate a theoretical LoD for a given setup, we must calculate the number of DNA copies per mL (x) that yields an average number of DNA copies per PCR well (k) that will result in a minimum of x positive PCR reactions of the total number of n PCR reactions. Where x of n (x/n) is the selected criteria for a positive test result. This situation can be described by the Binomial distribution. If we let the probability of a positive PCR be determined by the Poisson distribution, we can define a stochastic variable (X2) as the number of positive PCR reactions of the total number of PCR reactions, predicting the probability of an outcome of a minimum of x positive PCRs of a total of n PCRs with an estimated average of k copies per PCR well:
PðX2 xÞ ¼
n X n ð1 ek Þk ðek Þnk k
(3)
k¼x
Using [2], it follows that
Calculations of detection probability A mathematical prediction model was created to assess the sensitivity of a singleplex setup for detecting low-level DNA. On the basis of the average number of DNA copies per PCR well, a calculation of the LoD is possible. The model includes a calculation of the detection probability
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PðX2 xÞ ¼
n X n ð1 eux Þk ðeux Þnk k
(4)
k¼x
Probabilities of positive detection can be calculated for increasing k, and curves can be generated of accumulating © 2015 APMIS. Published by John Wiley & Sons Ltd
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detection probabilities, thus, ultimately generating a measurement of the probability of detecting the DNA in a positive sample with low-level DNA. We defined the 95% LoD (LoD0.95) as the number of DNA copies at which a 95% probability of detection is reached (expressed either as x copies per mL or k copies per PCR), with a given set of criteria for positive (x/n). This uniform model is applicable despite differences in method setup. The model can be set up in Microsoft Excel to calculate P, thus, estimating the chance of a positive result under different conditions and at different levels of DNA. See Table S1, for an Excel setup. We used MATLABÒ (version 7.14, The MathWorks Inc., Natick, Massachusetts, USA) to isolate x, given any value of P, x, n, and φ. We designed a script in MATLABÒ based on an F-solver. In addition, we aimed to find an approximation formula as a tool for a quick and easy evaluation of a given setup. For these investigations, we used curve fitting on groupings of k values related to (n/x) or n using Microsoft Excel 2007. The approximation formulas were tested against model data (using [3]) with Spearman Correlation, and Wilcoxon’s matched pairs signed rank test where a p value less than 0.05 was considered significant. Statistical calculations were performed using GraphPad Prism 5.02 (GraphPad Software, Inc., San Diego, CA).
DNA analysis To test the model-based in silico predictions of LoDs, we analysed repeated detection of low-level DNA in an average concentration of 0.5–3 DNA copies per PCR reaction and compared the outcome to model-based predictions for selected setups of x/n. We used the detection of diluted cffDNA as our test system. CffDNA was extracted from plasma from RhD negative pregnant women in gestational week 25 using the automated QIAsymphony SP extraction system (Qiagen, Basel, Switzerland) using the QIAsymphony DSP Virus/Pathogen Midi Kit using carrier RNA, and eluted in AE buffer. The PCR assay targeted the RhD gene, RHD (accession number NT_004610.19). Extracted RHD positive cffDNA was pooled and stored at 20 °C until analysis. The pool was diluted 1:10, and the cffDNA concentration was estimated using a five-point standard curve comprising reactions of 10 ng, 2 ng, 400 pg, 80 pg, and 16 pg DNA per PCR from an RHD hemizygous individual, detected by an RHD exon 7 assay (11). Then new dilutions of the pool were made with DNA concentrations equivalent to an average of 0.5, 1, 2, or 3 RHD positive DNA copies per PCR well. Dilutions were re-quantified by counting negative PCR reactions (N0) of a total (NT) of 96 reactions. The average number of copies per PCR reaction was calculated via k = ln(N0/NT) (13). Theoretical data were then calculated by the model based on this estimated k. A new real-time PCR analysis of the same dilution of DNA in 96 reactions was then carried out to obtain empirical data on the estimated k. Comparison of theoretical and empirical data was performed using Wilcoxon’s matched pairs signed rank test where a p value less than 0.05 was considered significant. The QIAgility instrument (Qiagen Inc.) was used for automatic PCR dispensing into MicroAmpÒ Optical 96-Well Reaction Plates (Applied BioSystems,
© 2015 APMIS. Published by John Wiley & Sons Ltd
Foster City, CA, USA). Real-time PCR was conducted using an ABI 7500 detection system (Applied BioSystems) with TaqMan chemistry. PCR for detection of RHD exon 7 was set up with 10 lL template volume in a total volume of 25 lL with Universal Master Mix. Final primer concentration was 900 nM and final concentration of the FAM- labelled probe was 100 nM. Primers and probe were synthesized by Eurofins MWG Operon (Edersberg, Germany). PCR profile was as previously described (11). Results were evaluated with the 7500 System SDS software version 1.4 from Applied BioSystems using auto baseline and a fixed threshold of 0.2. A PCR was considered positive if the cycle threshold (CT) value was below 42. Standard curve characteristics for the exon 7 assay were: y = 3.397x + 42.13, R2 = 0.998; PCR efficiency was 0.97.
RESULTS Detection probability
We developed a mathematical prediction model based on a combination of the Poisson distribution and the Binomial distribution. With this model, we were able to calculate LoDs for a range of scenarios to investigate the significance of the plasma equivalent per PCR and different applications of replicate PCR reactions and criteria for positive results in assays intended for detection of low-level DNA. We showed that increasing the plasma equivalent per PCR by a given factor improves the LoD0.95 by the same factor, see Fig. 1. In Fig. 2, the plasma equivalent per PCR is fixed at 0.1 mL showing different setup examples with different numbers of replicates and criteria for a positive result. It shows that if the criteria for a positive result allow at least one reaction to be negative (x < n), this will generate a lower LoD0.95 compared to a setup where all reactions must be positive (x = n), see Fig. 2. The PCR setups with different criteria for a positive result (x/n) were ranked by their sensitivity, calculated as the minimum average number of DNA copies per PCR resulting in a 95% probability of detection of a positive sample (k0.95), shown in Fig. 3. In addition, we found that an approximation calculation could serve as a quick calculation to guide choices of setup, see Fig. 4. For x < n and n < 8, the approximation of k0.95 was k0:95 ¼ 3:12ðn=xÞ1:15
(5)
with R2 = 0.969. From this approximation, we simplified the formula even further: k0:95 ¼ 3x=n
(6)
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Fig. 1. Predicting limits of detection (LoD) for three different plasma equivalents per PCR (50, 100 fixed criterion of at least 2/3 positive PCR reactions for a positive result. The LoD was defined as copies per mL resulting in a predicted 95%-probability of detection. The LoD was 40 DNA copies plasma equivalent per PCR of 50 lL. The LoD was 20 DNA copies for 100 lL, and 10 DNA copies
and 200 lL), using a the number of DNA per mL plasma for a for 200 lL.
Fig. 2. Predicting limits of detection (LoD) for different setup concerning replicate PCR reactions and criteria for positive result. The plasma equivalent per PCR was fixed at 100 lL. The LoD was defined as the number of DNA copies per mL resulting in a predicted 95%-probability of detection for each setup. Using 3/3 positive PCR reactions for a positive result yielded a LoD of 41 DNA copies per mL plasma (with a plasma equivalent per PCR of 100 lL). The LoD was 37 DNA copies for a criterion of 2/2, and the LoD was 20 DNA copies for at least 2/3, and 14 DNA copies for at least 2/4.
Using this simplified k, an approximated LoD0.95 in copies per mL (x0.95) could be calculated via [2] as: x0:95 ¼ 3x=nu
(7)
For x = n, k0:95 ¼ 0:99lnðxÞ þ 2:99 with R2 = 1. 734
(8)
A simplified approximation yielded an LoD0.95 in copies per mL as: x0:95 ¼ ðlnðxÞ þ 3Þ=u
(9)
Values of x0.95 obtained with different setups of x/n were calculated with all methods and tested for significant difference. For x = n, data from the simplified approximation were significantly different from model data generated in Microsoft Excel (Wilcoxon’s matched pairs signed rank test, p < 0.0001; © 2015 APMIS. Published by John Wiley & Sons Ltd
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Fig. 3. Different real-time PCR setups ranked according to the minimum average number of DNA copies per PCR at which a 95% probability of detection was reached (k0.95). Six is the maximum of total reactions presented.
Empirical test of model
Fig. 4. Curve fitting was applied to identify a simple relationship between x/n and k. Approximation: Fitted curve with corresponding curve statistics. Simplified approximation: The simplified approximation is shown in bold grey. Note that y = 3x1 is equal to k = 3x/n.
Spearman rs = 1.0000; n = 25) due to a systematic higher estimate of DNA copies, within 0.5 copies. With x < n at n < 8 there was no significant difference between model data generated from MATLABÒ and Microsoft Excel (Wilcoxon’s matched pairs signed rank test, p = 0.68; Spearman rs = 1.0000; n = 25) and no significant difference between model data generated from MATLABÒ and from the simplified approximation for x < n (Wilcoxon’s matched pairs signed rank test, p = 0.11; Spearman rs = 0.9935; n = 22). © 2015 APMIS. Published by John Wiley & Sons Ltd
Dilutions of low-level DNA were tested with realtime PCR, and the results were compared to model data generated in Microsoft Excel. With this experiment, we investigated whether empirical data obtained from laboratory testing would be in proximity of the data predicted by the model. Thus, given an estimate of k, we calculated a theoretical percentage (and a 95% confidence interval) of possible sets of replicate PCRs that would be positive and compared this with the observed percentage. We looked at assay setups with criteria for positive (x/n) of x/2 or x/3. All observed data fell within the 95% confidence interval, see Fig. 5. Overall, there was no statistically significant difference between the observed and the predicted frequencies of positive results (Wilcoxon’s matched pairs signed rank test, p = 0.8438; Spearman rs = 1.0000; n = 6). DISCUSSION We present a Poisson-based mathematical model for theoretical evaluation of singleplex setup choices for real-time PCR intended for reliable detection of low-level DNA, prior to standard laboratory optimization and validation procedures. Parameters that can be evaluated are the number of replicate PCR reactions, criteria for a positive result, and the plasma equivalent per PCR. In addition, we present a quick and easy approximation formula for rapid
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A
B
Fig. 5. Model predictions of outcomes of positive results with different criteria for positive (x/n) compared with empirical data obtained through PCR analysis, shown in percent. (A) Setups of 1/2 and 2/2 were analysed at k = 0.8, k = 1.8, and k = 3 DNA copies per PCR. (B) Setups of 1/3, 2/3, and 3/3 were analysed at k = 0.7, k = 1.8, and k = 3.5 DNA copies per PCR. Model predictions are presented with error bars representing 95% confidence intervals. Empirical PCR data are designated as observed. One experiment was carried out for each level of k, thus, representing a total of six experiments. Each experiment consisted of two PCR run, one for quantification of k, and, subsequently, one for analysis of predicted outcomes of x/n.
evaluation of setups for x < n (as well as a formula for x = n). For a new assay design, the consequence of different choices can be explored before initiating experimental work. Overall, these choices are central to the analytical sensitivity, and hence the diagnostic sensitivity and robustness of any given realtime PCR assay, in addition to other, well-known factors influencing the PCR assay. We based the model on the Poisson distribution of low-level DNA in a sample and the Binomial distribution of the PCR outcome as either positive or negative. In combination, this enabled us to describe the probability of obtaining a defined minimum number of positive PCRs based on the number of total PCR wells and the knowledge of the average number of DNA copies available in the sample. Importantly, the model allowed for the calculation of LoDs below an average of three copies per PCR. We presented calculations and experiments to demonstrate the applicability of the model. Based solely on mathematical work, it should be noted that the model does not aim to take into account certain physicochemical variables such as type of polymerase, inhibition from substances from the DNA extraction, or variations due to extraction technique. The model simply assesses the consequences of selected setup choices. In addition, a central assumption is that every present DNA copy is amplified, i.e. that we are in the range of singlecopy detection. This assumption is often fulfilled in practice. The technique is capable of detecting a single DNA copy (7, 16, 17). And as the number of positive reactions follows the Poisson distribution (10–12,18), and each copy is counted in the Poisson distribution, this implies directly that each copy will be amplified in the real detection system. 736
To investigate the applicability of our mathematical model, we tested the model-based predictions against empirical data obtained at 0.7–3.5 DNA copies per PCR. Exemplified by a selected number of setups, we demonstrated that the model predicts outcomes that are equal to those observed through experimental detection of low-level DNA. It shows that the model is solid enough for predicting realistic probabilities for outcomes of x/n given a certain known average level of DNA per PCR (k). With a k below 5, the distribution will match a Poisson distribution (13). Therefore, we could apply the k = –ln(N0/NT) formula, derived from Poisson, for quantification. Consequently, these experiments represented the testing of our model’s predictions in relation to experimental outcomes only under Poisson conditions. The model can be set up in Microsoft Excel or in, e.g. MATLAB. As an alternative, we developed a very simplified approximation formula of x0.95 = 3x/nφ (for x < n) that provides a quick assessment of any of the included parameters. The simple approximation formula was shown to fit the models from Microsoft Excel and MATLAB when n was kept below 8. Thus, the formula is simplified and limited; yet the applicability should be highly useful. In most cases, assay sensitivity can be calculated by head. See Box 1 for examples of how to use the formula. Using the formula to guide an assay setup should be combined with other considerations. For example, it would not be a good idea to use only one positive PCR reaction of many replicates as the criterion for a positive test outcome, as false positive reactions may occur, and such reactions would complicate the interpretation of the results. Overall, assessing the mathematical sensitivity via this formula may prevent wasting © 2015 APMIS. Published by John Wiley & Sons Ltd
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Box 1. Examples of how to use the simplified approximation formula
x0:95 ¼ 3x=nu We developed the approximation formula as a tool to enable a quick evaluation and design of PCR setups that are intended for reliable detection of low-level DNA. Abbreviations x0.95: estimated limit of detection (LoD) of DNA copies per mL sample with positive test outcomes in 95% of the cases; x: the number of positive PCR reactions; n = the number of total PCR reactions; x/n = the criterion for a positive test outcome; φ = the plasma-equivalent per PCR (mL), (plasma vol 9 template vol/elution vol). Examples of how to use the formula: Design a setup to detect one DNA copy per mL Aiming to detect only one DNA copy is an extreme example but demonstrates the use of the formula. Different setups can be used. If DNA is extracted from 5 mL, using an elution volume of 50 lL, and a template volume of 15 lL, then the plasma-equivalent per PCR, φ = (5 9 0.015/0.05) = 1.5. Then 1 = (3x)/(n 9 1.5) ⇒ (1.5/3) = (x/n) ⇒ 1/2 = x/n. Thus, the PCR setup could be two PCR reactions, with one positive PCR reaction as the criterion for a positive test outcome. Alternatively, if the DNA extraction is carried out from only 1 mL, using 50 lL elution, and a template volume of 10 lL, then φ = (1 9 0.01/0.05) = 0.2. Then 1 = (3x)/(n 9 0.2) ⇒ (0.2/3) = (x/n) ⇒ 1/6 = x/n. (This may, however, not be a wise setup due to the risk of false positive reactions interferring with the interpretation of the result). Find the LoD of a given assay setup Already known about the assay setup: DNA extraction volume = 1 mL, elution volume = 50 lL, template volume = 5 lL, x = 2, and n = 4. Calculations: The plasma-equivalent per PCR, φ = 1 9 0.005/0.05 = 0.1 mL. The LoD for the assay setup, x0.95 = (3 9 2)/(4 9 0.1) = 60/4 = 15 copies per mL. (This is an approximation of the exact LoD of 13.9 copies per mL). Design a setup when a parameter is fixed For practical or economical reasons, a setup may be limited by different requirements causing one or more parameters to be fixed at certain values. In general, the nonfixed values can be adjusted to improve the LoD, see below. Choose the number of PCR replicates If a LoD of 10 copies per mL is required and the plasma-equivalent is fixed at 0.2 mL, how many PCR replicates should be chosen to reach the required LoD? The following can be calculated: 10 = (3x)/(n 9 0.2) ⇒ 2 = 3x/n ⇔ 2/3 = x/n. Thus, one setup could be x = 2 and n = 3 as the criterion for a positive test outcome, i.e. at least 2 positive of 3 PCR replicates (the exact LoD is 10 copies per mL). Alternatively, one could choose 4 positive of 6 PCR reactions (the exact LoD is 9.38 copies per mL). Find the best plasma-equivalent per PCR If the PCR setup is fixed with 3 PCR reactions, and the criterion for a positive test outcome is that 2 PCR reactions must be positive, one can adjust the plasma-equivalent per PCR. Here, we can make different choices to secure a suitable LoD. We can calculate the following: x0.95 = (3 9 2)/(3 9 φ) ⇒ x0.95 = 2/φ. Thus, a plasma-equivalent per PCR of 0.2 would give a LoD of 10 copies per mL. If practically possible, we can choose an extraction volume of 2 mL, an elution volume of 50 lL, and a template volume of 5 lL, giving φ = 0.2. Other combinations of extraction, elution, and template volumes can also be used.
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Box 1 Continued
If x = n If the criterion for a positive test outcome is x = n, the LoD can be calculated using the formula: x0.95 = (ln(x) + 3)/φ. If, for example, the criterion is 2 positive of 2 PCR reactions and the plasma-equivalent per PCR is 0.15 mL, then the LoD is: x0.95 = (ln(2) + 3)/φ ⇒ x0.95 = (3.693/0.15) = 24.6 copies per mL.
valuable time in the laboratory. Assay adjustments can be performed to obtain a required LoD, before proceeding with standard laboratory work. For x = n, the statistically significant difference observed between the simplified model and the model in Microsoft Excel was due to a systematic higher DNA copy estimate, but only within 0.5 copies. This imprecision is of no practical consequence and acceptable for a simplified model. For example, if x = n = 3, then k0.95 = ln(3)+3 = 4.1 according to our simple model, corresponding to a LoD0.95 of 4.19 which was reported for the same setup (16). The ranking of k0.95 in Fig. 3 demonstrated the limit of three copies per PCR as the LoD for a setup where all replicates are demanded positive (x = n). This is in line with the observations from previously described theoretical limits for qPCR (8, 9). With our simple model for x = n, it is evident that k = 3 for n = 1, as ln(1) = 0. As seen for x = n in Fig. 3, k will increase as n increases. We presented our model for singleplex assay setups. It will be of great value to develop a similar model for multiplex setups. The possibility of evaluating the reliability of a given PCR setup is very important and may result in improved experimental design and increased speed to obtain a robust and accurate PCR-based assay. For testing of cffDNA, this is very important because the low quantities of cffDNA may hamper DNA detection especially in first-trimester analysis (6, 19). In addition, a high degree of variation and fluctuation (20) may also cause low levels later in pregnancy, so it is central that an assay setup is thoroughly evaluated and optimized, in the first as well as the second trimester of pregnancy. For everyone engaged in PCR analysis of low-level DNA, especially in a clinical setting, we recommend using our model to evaluate the mathematically determined assay sensitivity by estimating the LoD, or to include the model as a sensitivity assessment tool during design of new real-time PCR assays prior to extensive laboratory analysis. We recommend using the model as a tool to revise a PCR setup in general to improve or simplify the assay, or to revise the setup from an economical point of view. In our laboratory, the formula was 738
crucial in developing the routine antenatal assay to detect foetal RHD in maternal plasma, currently used in the Copenhagen region of Denmark (21–23). The track record of the assay demonstrates that the assay is extremely robust and highly sensitive, with only one false negative result in about 10,000 samples tested. In addition to the mathematical issues dealt with in this study, several other factors are central to the assay sensitivity of real-time PCR-based analysis of low-level DNA. For example, reliable analysis of cffDNA is affected by pre-analytical handling and sample processing (23–27), PCR-amplicon size (28, 29), DNA target repetition (12, 30, 31), and perhaps even nucleosome positioning (32). In conclusion, we have presented data and a mathematical description with formulas that enable easy evaluation of several important aspects of realtime PCR-based assays. The mathematical tool can help scientists to decide a specific setup for detection of low-level DNA by quickly evaluating a number of considerations that need to be taken into account before developing a new PCR based assay and prior to extensive laboratory optimization and validation procedures. Our simple mathematical model may assist laboratories in improving their assay sensitivity, facilitating a robust detection. CONFLICT OF INTEREST None declared. The authors acknowledge the contribution by Lone Vandsø Jørgensen and the staff from The Laboratory of Blood Genetics, Department of Clinical Immunology, Copenhagen University Hospital.
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RELIABLE DETECTION OF LOW-LEVEL DNA
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SUPPORTING INFORMATION Additional Supporting Information may be found in the online version of this article: Table S1. Calculate the likelihood (P) of a positive test outcome for a given set of parameters.
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