 # How to analyze  Dynamic Ranges of Detection & Quantification

First of all, let’s assume that a target gene concentration has been estimated together with its relative uncertainty, by applying the Poisson law as explained in the item [Poisson Law application].

Depending on the target concentration value in the well, the Relative Uncertainty forms a U-Curve which is made of 2 asymptotic curves:

• The sampling error curve, occurring at low target concentrations (on the left side in the [Figure A] below);

• The quantification error curve, occurring at high target concentrations (on the right side in the [Figure A] below). Figure A. Relative uncertainty curve (U-Curve) as a function of the target concentration
in a well that includes 28 000 partitions (in logarithmic view).

• The sampling error can be approximated as follows:

$$CU_{sampling} \sim \dfrac{z_c}{\sqrt{C_{0} \ V} }$$

where $$V$$ is the total partition volume, $$C_{0}$$ is the well concentration, and $$z_c = 1.96$$ for a confidence level of 95%.

• Note that, whatever the total number of partitions $$N$$ and the confidence level $$z_c$$ the relative uncertainty is always minimized at $$\dfrac{p}{N} \sim 79.7$$, which corresponds to the white point lying on the U-Curve in the[Figure 1] above.

The Minimal & Maximal Limits of Detection are represented by 2 vertical lines framing the U-Curve:

• The Minimal Limit of Detection $$LOD_{min}$$ at 95% (left vertical dotted line in the [Figure 1] above) is the smallest target concentration in the well for which probability of having at least 1 positive partition is more than 95%. If the Limit of Blank is zero, we have:

$$LOD_{min} = \dfrac{3}{V}$$

• The Maximal Limit of Detection $$LOD_{max}$$  at 95% (right vertical dotted line in the [Figure A] above) is the highest target concentration in the well for which probability of having at least 1 negative partition is more than 95%) for $$N$$ ranging from 10 000 to 1 000 000:

$$LOD_{max} \sim 10 \ \dfrac{N}{V}$$

• The Dynamic range of Detection $$DrD$$ at 95% is the number of decades from $$LOD_{min}$$ to $$LOD_{max}$$ at 95%. If the Limit of Blank is zero, we have:

$$DrD \sim log_{10}(N) + 0.5$$

• The Minimal & Maximal Limits of Quantification are defined with respect to a maximal relative uncertainty $$U_{max}$$ which is considered as acceptable for the current digital PCR experiment:
• The Minimal Limit of Quantification $$LOQ_{min}$$ at 95% is the smallest target concentration in the well for which the relative uncertainty at 95% is smaller than $$U_{max}$$ .Its value is given by the first intersection point of the 95% U-Curve and the horizontal line $$y = U_{max}$$ (which corresponds to the blue point lying on the U-Curve in the [Figure 1] above).

• The Maximal Limit of Quantification $$LOQ_{max}$$ at 95% is the highest target concentration in the well for which the relative uncertainty at 95% is larger than $$U_{max}$$Its value is given by the second intersection point of the 95% U-Curve and the horizontal line $$y = U_{max}$$ (see the [Figure 1] above).

• Finally, the Dynamic range of Quantification $$DrQ$$ associated with an acceptable uncertainty value $$U_{max}$$ is given by this formula:

$$DrQ (U_{max} )= LOQ_{max} (U_{max} )- LOQ_{min} (U_{max} )$$