**4. Analysis and evaluation of cdNAD**

This part focuses on the analysis and evaluation of cdNAD. The contents involve four aspects, which are the dehydration of microreaction, the dynamic range, the response of cdNAD, and the precision of cdNAD.

#### **4.1. Dehydration of microreaction**

In PCR, dehydration is attributable to the repeated denaturing steps, whereas, for isothermal NA amplification, reacting at a consistent temperature (e.g., 63°C) for at least 1 h results in dehydration. Also, the influence of dehydration is different according to the different materialbased chips for cdNAD.

As we know, the microfluidic PDMS chips suffer from dehydration heavily. Actually, the dehydration degree is determined by the thickness of the PDMS layer between the top of the chamber and the waterproof layer. If *v <sup>f</sup>* is the total volume fraction of reaction reagent, *v <sup>f</sup>* is defined by the function of

$$\nu\_f = \frac{\mathbf{A}\_{\text{chsmter}} \cdot \mathbf{h}\_{\text{chsmter}}}{\mathbf{A}\_{\text{PDMS}} \cdot \mathbf{h}\_{\text{PDMS}}}$$

where *A* and *h* refer to the designed area and height, respectively. Using the formula of *C sat\_25°C*×*P vap\_70°C*/*P vap\_25°C*, a saturated concentration of water vapor in PDMS at 70°C (*C sat\_70°C*) can be calculated as 400 mol/m3 . Then, the maximum fractional loss of water (*fl max*) from the reaction chambers is defined by

where *ρwater* is the density of water and *M water* is the molar mass of water.

#### **4.2. Dynamic range**

When high concentration targets are loaded, the chip panel can be completely saturated, whereas, for low targets, the chip panel still possesses the capability of realizing digital detection. Therefore, the theoretical dynamic range is determined by the high concentrations that make the chip completely saturated.

In this situation, the occurrence of an empty chamber is a small probability event, and due to statistical independence between the chamber number and the total empty chamber number, the event can be modeled as a random Poisson process. Then, the occurrence probability of *t* empty chamber number, *P (n=t, λ)*, is defined by *P*(*<sup>n</sup>* <sup>=</sup>*t*, *<sup>λ</sup>*)= *<sup>λ</sup> <sup>t</sup> e* <sup>−</sup>*<sup>λ</sup> t* ! . In the function, λ equals to the mean number of empty chambers in panels. For each empty chamber in each panel, namely, a chamber containing 0 molecule, the probability P(n=0,λ)=e-λ=e-m/N, where λ is the ratio of the loaded molecule number (m) to the chamber numbers (N). Therefore, λ is equal to the product of the probability P(n=0,λ) and the N chamber numbers, namely, λ=Ne-m/N. Finally, *P*(*n* =*t*, *λ*)=*P*(*n* =*t*, *N e* <sup>−</sup>*m*/*<sup>N</sup>* ) =(*N e* − *m N* ) *t e* <sup>−</sup>*<sup>N</sup> <sup>e</sup>* <sup>−</sup>*m*/*<sup>N</sup>* / *t* !. If λ is more than 10, the probability of the chip panel is completely saturated, P(n=0,λ)=e-10, which is less than 10-4 and can be as the accept‐ able failure rate. Then, λ=N e-m/N>10, and the maximum loaded molecular number (mmax) can be calculated.

#### **4.3. Response of cdNAD**

**4. Analysis and evaluation of cdNAD**

the precision of cdNAD.

140 Lab-on-a-Chip Fabrication and Application

based chips for cdNAD.

defined by the function of

be calculated as 400 mol/m3

chambers is defined by

**4.2. Dynamic range**

that make the chip completely saturated.

**4.1. Dehydration of microreaction**

This part focuses on the analysis and evaluation of cdNAD. The contents involve four aspects, which are the dehydration of microreaction, the dynamic range, the response of cdNAD, and

In PCR, dehydration is attributable to the repeated denaturing steps, whereas, for isothermal NA amplification, reacting at a consistent temperature (e.g., 63°C) for at least 1 h results in dehydration. Also, the influence of dehydration is different according to the different material-

As we know, the microfluidic PDMS chips suffer from dehydration heavily. Actually, the dehydration degree is determined by the thickness of the PDMS layer between the top of the chamber and the waterproof layer. If *v <sup>f</sup>* is the total volume fraction of reaction reagent, *v <sup>f</sup>*

> *chamber chamber <sup>f</sup> PDMS PDMS*

where *A* and *h* refer to the designed area and height, respectively. Using the formula of *C sat\_25°C*×*P vap\_70°C*/*P vap\_25°C*, a saturated concentration of water vapor in PDMS at 70°C (*C sat\_70°C*) can

When high concentration targets are loaded, the chip panel can be completely saturated, whereas, for low targets, the chip panel still possesses the capability of realizing digital detection. Therefore, the theoretical dynamic range is determined by the high concentrations

In this situation, the occurrence of an empty chamber is a small probability event, and due to statistical independence between the chamber number and the total empty chamber number, the event can be modeled as a random Poisson process. Then, the occurrence probability of *t*

the mean number of empty chambers in panels. For each empty chamber in each panel, namely, a chamber containing 0 molecule, the probability P(n=0,λ)=e-λ=e-m/N, where λ is the ratio of the loaded molecule number (m) to the chamber numbers (N). Therefore, λ is equal to the product of the probability P(n=0,λ) and the N chamber numbers, namely, λ=Ne-m/N. Finally,

where *ρwater* is the density of water and *M water* is the molar mass of water.

empty chamber number, *P (n=t, λ)*, is defined by *P*(*<sup>n</sup>* <sup>=</sup>*t*, *<sup>λ</sup>*)= *<sup>λ</sup> <sup>t</sup>*

. Then, the maximum fractional loss of water (*fl max*) from the reaction

*e* <sup>−</sup>*<sup>λ</sup> t* !

. In the function, λ equals to

*A h <sup>v</sup> A h* <sup>×</sup> <sup>=</sup> <sup>×</sup>

is

According to the distribution of molecules across the chip, the Poisson distribution is adapted to calculate the original concentration of target stock solution.

For each chip panel, the probability (*P*) of having the number (*n*) of NA molecules per chamber is *P*(*n,λ*)=(*λ <sup>n</sup> e* -*<sup>λ</sup>*)/*n*!, where *λ* is the ratio of loaded molecule number to chamber number of each panel. For *λ*, it is equal to the average molecule number per chamber, which can be expressed by the equation of *λ*=*C <sup>0</sup> X dil V*, where *C <sup>0</sup>* is the target's original concentration, *V* is the volume of each chamber, and *X dil* is the dilution factor of diluted targets used in the panel. If the stock solution is diluted in *k* fold, the *X dil*=*1*/*k*, and particularly, the *X dil* of stock solution is 1. When a chamber contains 0 molecule, the probability *P*(*n*=*0,λ*)=*e* -*<sup>λ</sup>*. As for a chamber capturing one or more molecules, the probability *P*(*n*≧*1,λ*)=1-*P*(*n*=*0,λ*)=1-*e* -*<sup>λ</sup>*. After dPCR, a chamber with observed positive signal suggests that at least one target molecule is captured. Therefore, the ratio (*f*) of observed positive chamber number to chamber number of each panel, namely, the observed fraction of positive chambers, equals to the probability *P*(*n*≧*1,λ*), which is *f*=*P*(*n*≧1,*λ*)=1-*e* -*<sup>λ</sup>*. Then, *λ*=*ln*(1-*f*)=-*C <sup>0</sup> X dil V*, and a linear variation relationship is exhibited in terms of the regression curve equation between *ln*(1-*f*) and *X dil*, because *C0* and *V* are constant values. Then, the target's original concentration *C <sup>0</sup>* can be calculated based on the slope of the curve (-*C <sup>0</sup> V*).

In addition, because the Poisson distribution is a particular case of a binomial distribution, the distribution of the positive chamber number (*x*) in each panel is classified as the binomial distribution. The probability distribution *P*(*x*) is therefore influenced by the probability *p*=*P*(*n*≧1,*λ*)=1-*P*(*n*=0,*λ*)=1-*e* -*<sup>λ</sup>*. Based on the nature of binomial distribution, *x* is equal to the mean, namely, *x*=*Np*=*N*(1-*e* -*<sup>λ</sup>*) (*N*, the total chamber number in each panel). Then, the average molecule number per chamber, *λ*=*ln*{*N*/(*N*-*x*)}, through which the loaded molecule number (*N <sup>e</sup>*) for each panel can be estimated by the equation *N <sup>e</sup>*=*Nλ*=*Nln*{*N*/(*N*-*x*)}.

#### **4.4. Precision of cdNAD**

When the chamber number in each panel is very large, the Poisson distribution is approximate to normal distribution, which is also called Gaussian distribution. Use the parameters in the discussion of response and dynamic range above, and let *y* be a random variable representing the number of positive chambers that capture at least one molecule. Also, we can know that its mathematical expectation or mean *μ* is *N* (1-*e* -*<sup>λ</sup>*), and its variance *σ <sup>2</sup>* is *Ne* -*<sup>λ</sup>* (1-*e* -*<sup>λ</sup>*). Then, the probability density function associated with *y* is as follows:

If the precision of cdNAD is defined as the minimum difference in concentration (∆*λ*/*λ*) that is reliably detected with more than 99% true positive and more than 99% true negative rate, this situation corresponds to 4.6*σ* separation in the mean (*μ*) for two Gaussian distributions. That is to say, in the case with small ∆*λ*, ∆*μ*/∆*λ*=4.6*σ*. Then, ∂[*N*(1-*e* -*<sup>λ</sup>*)]∆*λ*/∂*λ*=*Ne* -*<sup>λ</sup>*∆*λ*=4.6[*Ne <sup>λ</sup>* (1-*e*-*<sup>λ</sup>*)]1/2, and ∆*λ*/*λ*=4.6(*e <sup>λ</sup>*-1)1/2/(*λN* 1/2). For different total chamber numbers (*N*) in chip panel, the precision (∆*λ*/*λ*) to expected molecules per chamber (*λ*) could be plotted.
