In the biological part, our device shows differential response to different amounts of 2,4-dinitrotoluene (DNT)
, which is manifested as a difference in the amount of fluorescent protein. Our ultimate goal is to use the expression of fluorescent protein to reflect the distance between the device and the landmine, and to approximately determine the area where the landmines are most likely located based on the positions of multiple devices and the expression of fluorescent protein. Then, we can conduct visual analysis and design a safe demining route. Therefore, in the modeling part, we establish two models to help us achieve this goal.
The first model: establish the mathematical relationship between DNT amounts and the distance from the landmine surface.
The second model: establish the mathematical relationship between fluorescent protein expression and DNT amounts.
In model 1, we use the model to simulate the relationship between the DNT amounts of the soil and the distance from the landmine.
1.1 The Diffusion Model Based on the Exudation Mechanism of the Explosives in the Landmine
In this model, according to previous reviews that studied and analyzed the mechanism of the seepage of explosive components in explosives , we establish a theoretical model of DNT diffusion in the soil.
1.1.1 Model Analysis and Establishment
According to the literature , 2,4,6-Trinitrotoluene (TNT), the main component of explosive molecules, can exude into the external environment from the interior of a landmine through the polymer materials. Thus, we obtain the relationship model between TNT amounts and the distance from the landmine at different time. If we consider the soil as a loosely structured polymer, we can obtain the corresponding model of the relationship between the amounts of DNT in the soil and the distance from the landmine.
We establish a system of differential equations to simulate the entire diffusion process:
Equation (1) expresses the diffusion process of DNT in the soil, which represents the relationship between the DNT amounts in the soil and the distance from the landmine over time.
Equation (2) is the distribution of DNT in the soil at the initial moment.
Equation (3) is the DNT amounts at x=0 (landmine surface), which is replaced by the initial DNT amounts on the landmine surface.
Equation (4) is the amount of DNT when x=d (anywhere), which is described by the double-membrane mass transfers theory. Here we only describe the calculation of the parameters, but do not describe this theory.
Fig. 2. DNT diffusion diagram.
C' is the amount of explosive molecules in the external environment.
D1 and D2 are the diffusion coefficients of explosive molecules in solid (soil) and gas (air) respectively.
δ1 and δ2 are the thickness of the solid phase film and the gas phase film respectively.
S1 and S2 are the solubility of explosives in gas and solid respectively.
Fig. 3. The double-membrane mass transfers theory.
1.1.2 Model Solution
According to the calculation example given in the literature , we can change the parameters and use the MATLAB to obtain the relationship between the DNT amounts and the distance at different time.
Therefore, the parameters that we need are as follows:
Tab. 1. Parameters of model 1.
The above model can be solved by using the MATLAB. We can get the image of the DNT amounts changing with time, and the image of the DNT amounts changing with the distance from the landmine. We set the time to the final stable time to get the image of the DNT amounts changing with the distance from the landmine at the stable time.
1.1.3 Results Analysis and Prospects
Since we are unable to find the relevant DNT diffusion coefficient parameters in soil and air after consulting many documents, we could not really bring it into the calculation. This complex model is very precise, and it has many computational parameters. Hopefully, further research will help us complete this model.
The model can be used for:
(1) Due to difference of landmine types, shell materials and manufacturing processes, the amount of DNT exudation may also vary. As long as the DNT amounts on the landmine surface C1 brought into different types of landmines, the model can simulate the relationship between the DNT amounts in the soil and the distance of different types of landmines.
(2) As long as there are relevant parameters, we can also assess the exudation models of the molecular components from other explosives in different polymer materials.
1.2 The Diffusion Model Based on the Adsorption Capacity of DNT in the Soil
1.2.1 Model Analysis and Establishment
Since the above model lacks parameters, we can simulate this process with a completely idealized simple model.
(1) DNT leaks evenly on the surface of a spherical landmine.
(2) The adsorption rate of soil to DNT remains constant.
(3) The weight of soil per cubic centimeter is 2.5 g.
Tab. 2. Parameters of model 1.
3. Model and its solution
According to the literature , when adsorption equilibrium occurs at around 20 ℃ after 24 h, the adsorption capacity of soil to DNT is 1.38 μg/g, that is, 1 g soil can adsorb 1.38 μg DNT, and it can be considered that the weight of soil per cubic centimeter is about 2.5 g. We assume that the radius of landmines is 1 cm.
1.2.2 Analysis of the Results
Use the MATLAB and get the following results:
When the amount of DNT is 10 g and the distance from the landmine is about 88 cm, the DNT amounts are close to 0 g.
Fig. 4. Relation between DNT amounts and distance from landmine.
1.3 Analysis of the Advantages and Disadvantages of the Model
The diffusion model based on the exudation mechanism of explosive components in the landmine can better solve the problem of DNT diffusion in the soil. However, due to the lack of parameters, we cannot use the limited research conditions to complete the research process.
As for the diffusion model based on the adsorption capacity of the DNT in the soil, through our ideal hypothesis, we have obtained the change of DNT amounts as the distance from the landmine increases, and obtained the critical value of various initial amounts of DNT.
Comparing the example given in the literature  with the image of the diffusion model of TNT in polymer, due to the similarity between the chemical properties of TNT and DNT, we find that when the initial amount of DNT is 52 μg, the graphic trend of the diffusion model based on the exudation mechanism of the explosives in the landmines is roughly similar to the graphic trend of the diffusion model based on the adsorption capacity of DNT in the soil, especially when the time is long enough. Therefore, it is proved that the diffusion model based on the adsorption capacity of DNT in the soil is representative.
2.1 The Data Processing
The data can be used to establish the relationship between DNT induced amounts and reporter proteins:
DNT induced fluorescence of DH5α/pYB1a-yqjF-EGFP, and EGFP fluorescence intensity is used as the reporter.
The data is the ratio of measured bioluminescent intensity versus OD600 every 30 min under different induction conditions. In a testing of 12 h, we totally collect 12 sets of data.
After subtracting the LB blank, the experimental group is calculated as follows:
Control: No induction group
Take the data of the 100th min of induction culture for comparison, and use the statistical software SPSS for curve regression analysis.
In a large number of regression analysis, the relationship between the two variables is linear, or can be transformed into linear correlation. However, there are also many nonlinear correlations. For example, in the linear motion with a constant velocity, the relationship between the distance and time is a quadratic function; The free-falling body motion and parabolic trajectory are all nonlinear. Curvilinear regression is to study the nonlinear relationship between a dependent variable and an independent variable, and we can generate a regression equation from it.
Curve regression in the SPSS has two requirements for the data.
We can only deal with the curve equation with only one independent variable.
We can only deal with the curve equation that satisfies the essential linear relation. The essence is the linear relationship, which means that the relationship between variables is nonlinear in form, but it can still be transformed into the linear relationship through data transformation.
Ten types of curves can be used in SPSS: quadratic curve, cubic curve, compound curve, growth curve, exponential curve, logarithmic curve, S curve, power curve, inverse function, and logic function. These types can basically meet the needs of conventional analysis. The following table lists the main curve types and their expressions:
Tab. 3. Parameters of model 2.
2.3 Curves Regression Steps
2.3.1 Judgment of Curves Types Based on Scatter Graph
In order to reduce the blindness of curve estimation, a scatter plot is usually used to evaluate the relationship between independent variables and dependent variables to determine whether there is a clear logical relationship between them. If there are several scatter points near the curve in a scatter plot, which are close to one curve, the curve regression analysis can be made according to the preliminary judgment; Otherwise the curve estimation cannot be processed. For the data that can be used for curve estimation, the shape of the curves should be analyzed carefully to determine which kind of curve it belongs to, including parabola, logarithmic curve and exponential curve.
2.3.2 Regression Analysis of Execution Curve
The process includes creating a curve estimation function, setting dependent variable and independent variable correctly in the configuration interface of "curve estimation", and selecting several curve types at the same time. After completing the computer processing of curve regression, the most appropriate curve types are selected according to the output results, referring to the R2 value of the judgment coefficient and the sig value of the test probability.
Finally, according to the coefficient values of the curve type, the final function formula is written.
The chart is shown below:
Tab. 4. Model summary and parameter estimates.
Tab. 5. Model parameters for the data.
Fig. 5. Model curve fitting diagram of the data.
2.3.4 Summary of Custom Equations
According to the above two models, when we get the fluorescent protein expression level of the device, we can estimate the approximate DNT amounts of the site from Model 2, and then calculate the approximate distance between the site and the landmine according to Model 1. According to the results of multiple devices, a probabilistic image of the distribution of landmines in a region can be formed. Red represents the most probable landmines distributed positions, and the color gradually fades from red to blue, indicating that the probability of landmines distribution is gradually decreasing. Based on the image, we can design the landmine clearance route or lightning protection route with minimal casualties or injuries.
Through the models above, we can evaluate the approximate distance between the location of fluorescent protein expression response device and the surface of the landmine, and realize the visual analysis.
Fig. 6. The flow chart.
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 Donsovakm K.M.，Pennington J.C.，HAYES C., et al. Dissolution and transport of 2,4-DNT and 2,6-DNT from M1 propellant in soil[J] Chemosphere, 2009, 77(4):597-603.