# On the Determination of Uncertainty and Limit of Detection in Label-Free Biosensors

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## Abstract

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## 1. Introduction and Review of Basic Concepts

_{1}to concentration C

_{N}. n independent measurements are repeated in each concentration of the calibration function. From these measurements, we can obtain the mean values and standard deviations of the signal at each concentration point and finally, through the linear regression, the parameters a and b of the linear calibration function, where a is the slope of the straight line and b the intersection with the vertical axis:

_{i}measurements is carried out in the ith point of a calibration curve, we can compute estimations for μ and σ with the following expressions:

_{C}) of the signal y, is defined as the value, the exceeding of which leads, for a given error probability α, to the decision that the concentration is not zero [6,9,10,20] when measuring a measurand without analyte. The detection limit of the signal y (y

_{LoD}) is defined as the central value of a Gaussian distribution which probability of being below critical value y

_{C}is β. Error probabilities α and β are, respectively, the probabilities of false positive and false negative, and should be chosen according to the confidence level required. Figure 3 shows graphically the relations between these parameters for cases $\alpha \ne \beta $ and $\alpha =\beta $.

_{B}, y

_{C}, y

_{LoD}, α and β we indicate the following examples:

_{LoD}− y

_{C}= 1.645 σ, y

_{C}− y

_{B}= 1.645 σ and y

_{LoD}− y

_{B}= 3.29 σ

_{LoD}− y

_{B}= 3σ then α = β = 0.067 (6.7%). $k=3$ is the value recommended by [15].

_{LoD}− y

_{B}= 3σ and y

_{LoD}= y

_{C}, then the error probability α would be extremely small, less than 0.0015 (0.15%), but error probability β would be as great as 0.5 (50%).

_{B}and y

_{LoD}([8], page 839):

_{B}is the mean of the blank measurements, s

_{B}is the standard deviation of the blank measurements. The numerical factor k is chosen according to the confidence level desired. (See Figure 3 and its subsequent explanation).

_{LoD}), normally noted as LoD can be easily calculated from the analytical sensitivity a of a previously done calibration function or calibration curve:

## 2. Determination of Uncertainty in a Measuring Interval and the Limit of Detection through the Calibration Function Data

_{C}is the combined standard uncertainty for concentration that is estimated combining the standard uncertainties u

_{y}, u

_{b}and u

_{a}; we suppose a and b are the only correlated quantities; and r(a,b) is the estimated correlation coefficient for a and b quantities [21,24].Applying Equation (8) to Equation (7) we obtain:

_{a}, u

_{b}and r(a,b) are obtained from a set of n

_{i}signal measurements y

_{ij}{j = 1,.. n

_{i}} taken for N concentrations C

_{i}{i = 1,…N}. Assuming that signal measurements y

_{j}, for each concentration, have a normal distribution, they will be characterized by its means and standard deviations {${y}_{i},{s}_{{y}_{i}}$} (See Equations (3) and (4)). Additionally, if we assume negligible the uncertainty in the independent variable C

_{i}compared to the uncertainty of the signal y

_{i}(Equation (10)), we can obtain the parameters a and b from the N triads $\left\{{\mathrm{C}}_{\mathrm{i}},{\mathrm{y}}_{\mathrm{i}},{\mathrm{s}}_{{\mathrm{y}}_{\mathrm{i}}}\right\}$ and Equations (11)–(13) [25]:

**I**is the identity N × N matrix, and Equation (21) could be expressed as:

_{C}by a coverage factor k and defines an interval that should include a large fraction of the distribution of values that could reasonably be attributed to the concentration [21,24]. The value of k considered is 3 to have a level of confidence greater than 99%:

_{B}to the uncertainty of the signal for the zero concentration. If we assume that b is determined without uncertainty. Equation (28) becomes similar to the definition of IUPAC expressed in Equation (6). The difference is that the statistical estimator s

_{B}has been replaced by the standard uncertainty associated with the null concentration:

_{s}) and the resolution (u

_{R}). As indicated at the end of the previous section, there are other contributions whose analysis would lead us to more complex and less general models. For the statistical uncertainty is possible to use as the best estimator of the data dispersion, the standard deviation of the mean, which is obtained by dividing by $\sqrt{n}$ the standard deviation s

_{B}of the sample [24]:

_{R}can be calculated by the Equation (31), assuming a uniform distribution of probability between −R/2 and R/2 [21,24]:.

_{s}) and resolution (u

_{R}), are taken into account in the calculation of the C

_{LoD}of Equation (32). However, either u

_{R}, u

_{s}or u

_{b}could be negligible depending on the biosensing system analyzed.

## 3. Analysis of Standard Immunoassay Situations

#### 3.1. Simulated Immunoassay

#### 3.2. Experimental Immunoassay

- ○
- The biosensor response is non-linear, with a noticeable lack of repeatability at the end of its scale (over 20 μg/mL).
- ○
- Variability increases with concentration probably due to differences in sensing cells or biofuntionalization performance among other possible factors (standard deviations ${s}_{i}$ do not pass the Hartley’s test [29]).

- ○
- The degrees of freedom of the problem $=N-k$ , where $k$ is the number of the parameters to be determined during the fitting: five for the 5PL, $g+1$ for the polynomials.
- ○
- The weighted sum of squares $={{\displaystyle \sum}}_{j=1}^{N}{\left({y}_{ij}-{\widehat{y}}_{i}\right)}^{2}/{\widehat{s}}_{{y}_{i}}^{2}$, where ${\widehat{y}}_{i}$ is ${\widehat{y}}_{i}=f\left({C}_{i}\right)$. $f\left(C\right)$ is the fitted calibration curve.
- ○
- The critical value ${\chi}_{c}^{2}$ of a Chi-square distribution with $\nu =N-k$ degrees of freedom corresponding to a confidence level of $\beta =95\%$.
- ○
- $AICc=N\xb7log\left(Q/N\right)+2k+2k\left(k+1\right)/\left(N-k-1\right)$.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**(

**A**) Critical value, Limit of Detection, false positives and false negatives for the general case; (

**B**) Critical value, Limit of Detection, false positives and false negatives for the case α = β.

**Figure 5.**Theoretical sets of points representing an immunoassay and calibration function build for the first nine points.

**Figure 6.**Data, calibration line, uncertainty band, limits of detection and quantification and measuring intervals for the given example.

**Figure 9.**Data, calibration curve for parabolic fit, uncertainty band, limits of detection and quantification and measuring intervals for the given example.

**Figure 10.**Data from Table 4 and calibration curve for parabolic fit.

**Table 1.**Numerical values represented in Figure 5.

C (μg/mL) | 0 | 1 | 10 | 20 | 25 | 30 | 40 | 50 | 60 | 100 | 200 | 300 | 400 | 500 |

Signal (A.U.) | 0 | 1.9 | 17.7 | 32.3 | 38.5 | 44.2 | 54.1 | 62.2 | 68.9 | 85.7 | 97.9 | 99.0 | 99.1 | 99.2 |

S (A.U.) | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |

**Table 2.**Numerical values of the parameters of Figure 6.

a([A.U.]/[μg/mL] | b([A.U.]) | u_{a}([A.U.]/[μg/mL] | u_{b}([A.U.]) | r |

1.17 | 4.88 | 0.05 | 1.66 | −0.79 |

LoD (μg/mL) | LoQ (μg/mL) | U_{min} (μg/mL) | U_{max} (μg/mL) | C_{Max}(μg/mL) |

5.7 | 16.8 | 4.5 | 6.3 | 60 |

**Table 3.**Numerical values of calibration lines, uncertainty band, limits of detection and quantification and measuring intervals for the given example.

N = 9 | N = 8 | N = 7 | N = 6 | |
---|---|---|---|---|

a([A.U.]/[μg/mL] | 1.17 | 1.27 | 1.38 | 1.49 |

b([A.U.]) | 4.88 | 3.37 | 2.03 | 1.04 |

LoD (μg/mL) | 5.7 | 5.4 | 5.1 | 4.9 |

LoQ (μg/mL) | 17.1 | 16.2 | 15.3 | 14.7 |

C_{max}(μg/mL) | 60 | 50 | 40 | 30 |

U_{min} (μg/mL) | 4.5 | 4.3 | 4 | 3.8 |

U_{max} (μg/mL) | 6.3 | 5.9 | 5.6 | 5.1 |

**Table 4.**Numerical values from calibration of six BICELLs (Biophotonic Sensing Cells) for anti-IgG detection.

Concentration (μg/mL) | Transduction Signal ${y}_{\mathit{ij}}$ (nm) | Mean ${\overline{y}}_{i}$ (nm) | S_{i}(nm) | |||||
---|---|---|---|---|---|---|---|---|

j = 1 | j = 2 | j = 3 | j = 4 | j = 5 | j = 6 | |||

1 | 0.13 | 0.15 | 0.00 | 0.09 | 0.11 | 0.07 | 0.09 | 0.05 |

2.5 | 0.52 | 0.43 | 0.03 | 0.28 | 0.26 | 0.45 | 0.33 | 0.18 |

5 | 0.61 | 0.72 | 0.44 | 0.35 | 0.59 | 0.45 | 0.53 | 0.14 |

7.5 | 0.87 | 0.90 | 0.67 | 0.87 | 0.67 | 0.51 | 0.75 | 0.16 |

10 | 1.43 | 1.39 | 1.17 | 1.39 | 1.08 | 0.85 | 1.22 | 0.23 |

15 | 2.22 | 2.10 | 2.00 | 2.10 | 1.92 | 1.71 | 2.01 | 0.18 |

20 | 3.40 | 3.28 | 3.31 | 3.40 | 2.97 | 2.60 | 3.16 | 0.32 |

30 | 3.92 | 3.80 | 3.96 | 4.18 | 3.37 | 3.14 | 3.73 | 0.40 |

50 | 4.55 | 4.82 | 4.68 | 4.30 | 4.26 | 3.73 | 4.39 | 0.39 |

70 | 5.38 | 5.07 | 5.24 | 4.94 | 4.77 | 4.91 | 5.05 | 0.22 |

100 | 6.14 | 5.69 | 5.76 | 5.66 | 5.37 | 5.05 | 5.61 | 0.37 |

**Table 5.**Numerical values from calibration of six Biophotonic Sensing Cells (BICELLs) for anti-IgG detection.

Type of Curve | Deegres of Freedom $\mathit{\nu}$ | Weighted Sum of Squares $\mathit{Q}$ | Chi-Square Critical Value ${\mathit{\chi}}_{\mathit{c}}^{2}$ | Akaike Information Criterion AICc |
---|---|---|---|---|

Polynomial g = 1 | 5 | 37.1 | 11.1 | 18.7 |

Polynomial g = 2 | 4 | 8.66 | 9.49 | 15.5 |

Polynomial g = 3 | 3 | 6.31 | 7.81 | 27.3 |

Polynomial g = 1 | 2 | 4.16 | 5.99 | 66.4 |

5PL | 2 | 2.01 | 5.99 | 61.3 |

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**MDPI and ACS Style**

Lavín, Á.; Vicente, J.D.; Holgado, M.; Laguna, M.F.; Casquel, R.; Santamaría, B.; Maigler, M.V.; Hernández, A.L.; Ramírez, Y.
On the Determination of Uncertainty and Limit of Detection in Label-Free Biosensors. *Sensors* **2018**, *18*, 2038.
https://doi.org/10.3390/s18072038

**AMA Style**

Lavín Á, Vicente JD, Holgado M, Laguna MF, Casquel R, Santamaría B, Maigler MV, Hernández AL, Ramírez Y.
On the Determination of Uncertainty and Limit of Detection in Label-Free Biosensors. *Sensors*. 2018; 18(7):2038.
https://doi.org/10.3390/s18072038

**Chicago/Turabian Style**

Lavín, Álvaro, Jesús De Vicente, Miguel Holgado, María F. Laguna, Rafael Casquel, Beatriz Santamaría, María Victoria Maigler, Ana L. Hernández, and Yolanda Ramírez.
2018. "On the Determination of Uncertainty and Limit of Detection in Label-Free Biosensors" *Sensors* 18, no. 7: 2038.
https://doi.org/10.3390/s18072038