# Note on the PCR threshold standard curve

- Benedict G. Archer
^{1}Email author

**Received: **6 July 2017

**Accepted: **30 November 2017

**Published: **11 December 2017

## Abstract

### Objective

The PCR threshold standard curve is based on an exponential model of the initial phase of a PCR run where template replication efficiency is constant cycle to cycle. As such it requires that a threshold is at a level of amplified template not higher than where replication efficiency falls from its initial value. A second requirement is that all amplification profiles, both calibration and test, have the same initial efficiency. However, whether these requirements are met may not be checked, and there seems an apparent awareness that thresholds can be set higher than where efficiency has dropped from the initial value without compromising result validity. The objective of this study is to reconcile using the method without satisfying the requirement that amplification is exponential at threshold level.

### Results

Substituting the more general requirement that profile shapes be congruent to threshold level, except for translation along the cycle axis, and a derivation of the standard curve that includes cycles beyond the exponential phase accomplishes the objective without affecting usage of the method or any prior results and enables a practicable way to verify that the second requirement for same initial efficiency is satisfied.

## Keywords

## Introduction

The threshold standard curve method, regarded a gold standard for quantitative PCR, is based on an exponential model of the initial phase of a PCR run. The requirement that the threshold is at a level where amplification is exponential, that is, in the initial, constant efficiency phase of a run, is explicit in derivations of the straight line model of the threshold standard curve presented in articles, tutorials, commercial guides, etc. Many sources, e.g. [1], include the requirement in discussing the method, but some also downplay the importance of where a threshold level is set. There seems a tacit understanding, perhaps engendered by the great number of valid analyses where the requirement is not verified, or is even recognized to not be met, e.g. [2], that strict conformance is not critical. However, an explanation why calibrations and analyses using a threshold set at a level higher than where the exponential phase of the run has ended can be valid and a derivation of the model for the standard curve that avoids the requirement are lacking.

We show in this note that the validity of a standard curve, predicted target amounts in unknown samples based on it, the efficiency estimated from its slope and the linear relationship between \(C_T\) and the logarithm of target amount in a calibration sample do not depend on threshold level provided a more general requirement is met. We make a logical case for replacing the requirement regarding efficiency with a less restrictive requirement, and we derive the line modeling the standard curve in a way that includes cycles beyond the exponential (constant efficiency) phase of a run. Our model of the standard curve remains a straight line representation of \(C_T\) as a function of the logarithm of calibration target amount and does not change how a calibration is constructed or used to estimate unknown target amounts.

## Main text

### Analysis

Our analysis identifies a characteristic of amplification profiles implied by the requirement of constant efficiency but more general, substitutes it for requiring that thresholds are within the exponential phase, and extends the derivation of the threshold standard curve to incorporate the new requirement.

#### Requiring shape congruency instead of constant efficiency to threshold

Amplification profiles having the same initial efficiency and maintaining that efficiency to a threshold level will have the same shape to that threshold except for translation along the cycle axis. Two profiles with different initial efficiencies cannot be superimposed by shifting along the cycle axis. If a threshold is set higher than is reached within the exponential phase but at a level where the profiles have the same shape, differences in \(C_T\) from one profile to the next (and slopes of standard curves) will be the same as for any lower threshold. Thus substituting shape congruency to a threshold for the requirement that thresholds be set within the exponential phase also allows setting thresholds beyond the exponential phase.

#### Modifying the straight-line model of the standard curve

We briefly recapitulate the derivation of the equation for the standard curve in order to show how to incorporate the new requirement that extends the model to include cycles beyond the exponential phase.

#### Derivation of the standard curve for threshold within exponential phase

*k*by \(E_k\), amount of starting target DNA by \(X_0\), and a threshold amount of amplicon \(X_T\) corresponding to a number of cycles \(C_T\), any PCR run is represented by the general equation,

#### Derivation of the standard curve for threshold after exponential phase

### Discussion

We demonstrate the extended model using data published by Rutledge and Stewart [2] who in a study of methods for estimating efficiency presented data that are an example of the premise of this analysis. We also briefly note other insights afforded by the revised equation for the standard curve line.

#### Applying the new standard curve to data

Attributes of calibrations at various thresholds in Figure 2A of [2]

Threshold | \(E_1\) | \(X_T\left[ \frac{\left( 1+E_1\right) }{\left<1+E_\delta \right>}\right] ^\Delta\) | \(X_T\) | \(\left[ \frac{\left( 1+E_1\right) }{\left<1+E_\delta \right>}\right] ^\Delta\) | \(\Delta\) | \(E_\delta\) |
---|---|---|---|---|---|---|

500 | 0.926 | 1.75e+09 | 1.74e+09 | 1.00 | 0 | − |

1000 | 0.920 | 3.48e+09 | 3.49e+09 | 1.00 | 1 | 0.92 |

2000 | 0.925 | 7.67e+09 | 6.97e+09 | 1.11 | 2 | 0.83 |

3000 | 0.938 | 1.56e+10 | 1.05e+10 | 1.49 | 3 | 0.70 |

5000 | 0.930 | 3.22e+10 | 1.74e+10 | 1.85 | 4 | 0.65 |

7000 | 0.929 | 6.90e+10 | 2.44e+10 | 2.83 | 5 | 0.57 |

9000 | 0.911 | 1.68e+11 | 3.14e+10 | 5.35 | 7 | 0.50 |

The equivalence of \(E_\delta\) with \(E_1\) at the \(1000\,\) FU threshold level and the decrease in \(E_\delta\) at \(2000\,\) FU tell us that the constant efficiency portion of the run ended at an amplicon amount corresponding to a signal level between \(1000\,\) and \(2000\,\) FU, consistent with the analysis in [2]. The progressive decrease in \(E_\delta\) for higher thresholds shows the continuing decrease in efficiency after the \(2000\,\) FU threshold is passed but still several cycles before the abrupt decrease in efficiency signaling the onset of the plateau phase which was 10,000 FU or higher.

#### Implications for quantification

A standard curve constructed from a higher threshold is moved higher on the \(C_T\) vs. \(\log (X_0)\) plot as a result of threshold crossings occurring later, but the slope and efficiency estimated from the slope are not changed provided the profile shapes are the same to the threshold level. The caution for threshold standard curve quantification, that initial efficiency must be the same for all samples, both calibration and test, is included in the requirement that the shapes of all profiles be the same which can be alternately stated as requiring that the initial efficiency and fall off in efficiency that occurs as amplicon amount increases must be the same *to threshold level* for all samples. Profiles developed under the same conditions will typically conform to the new requirement some cycles beyond the end of the exponential phase but thresholds must still be set at levels where efficiency is not too low, e.g., \(> 50\%\), and before differences in profiles owing to sample effects, amplification of a second target, etc. develop.

Verifying that slopes of standard curves (or efficiencies derived from the slopes) constructed from thresholds at different levels are the same validates that the calibration data, up to the highest threshold tested, meets the requirements of the method. Verifying that the predicted target amount of a test sample is consistent when estimated from two or more calibration lines constructed at threshold settings where calibration profiles are shape-congruent validates that a test sample data meets the requirements of the method.

#### Application to methods using a standard curve not based on \(C_T\)

Requiring profiles to have the same shape would also apply to methods that instead of a \(C_T\) use the cycle position of a different profile attribute, e.g., a derivative maximum [3]. As an aside we note that because an exponential and all its derivatives are increasing functions, a derivative maximum is always beyond the exponential phase.

## Limitations

The revised model of the threshold standard curve removes the requirement that threshold crossings occur within the exponential phase of a run, provides a way to estimate where the exponential phase ends and enables new ways to check validity of both calibration and test sample data, but does not change how a standard curve is constructed or the calculation of unknown target amount in a sample. The limitations of the threshold method are not affected by the results of this study.

## Declarations

### Acknowledgements

Not applicable.

### Competing interests

The authors declare that they have no competing interests.

### Availability of data and materials

The manually de-plotted data from [2] is available in Additional file 1 submitted with manuscript.

### Consent for publication

Not applicable.

### Ethics approval and consent to participate

Not applicable.

### Funding

None.

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## Authors’ Affiliations

## References

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