Modeling a nd Simulation of Cancer Treatment Using Cold Atmospheric Plasma

The cold atmospheric plasma (CAP) becomes a promising technology for the cancer cell treatment. There are many aspects affecting the effect of the treatment including plasma discharge voltages, CAP exposure time, cancer cell type and so on. In order to have a further understanding the cancer treatment using CAP jet, we proposed a mathematical model by using the least square method for the response of cancer cell line of U-87 MG with CAP jets treatment based on experimental data from reference. The comparison demonstrates that the mathematical model can capture the characteristics of the cancer cell viability in the experimental data. It means that we can use the same method to predict cancer cell response to CAP under a nominal condition. We also proposed the mathematic model using the Taylor expansion method according to the processed data to study the correlation between the cell viability, the treatment and the CAP exposure time.

Kim et al applied the CAP jet for lung carcinoma treatment, demonstrating the effects of the CAP jet on the mouse lung carcinoma and fibroblast cells [17]. The CAP jet was applied in breast cancer treatment and the results showed that the CAP jets had a very sensitive effects on the breast cancer cells [18]. The plasma jet was investigated for the anti-cancer treatment, and the cell death effects of the CAP jet and its molecular mechanisms were studied using the N2 and air plasma jets, demonstrating the potential employment of the CAP jets in the cancer therapy. It is found that CAP jets can be used to generate the oxygen species to initiate the cancer cell apoptosis [19].
Graves et al presented a simulating model to analyze the thermal effects of the target using the CAP jet dynamics, showing the importance of the feedback control on the achieving spatially uniform dose delivery [20,21].
Researchers found that there are many plasma parameters affecting the cancer treatment including the applied voltage, temperature, properties of the target, cancer types, and plasma composition, and so on [12,13,15,16]. What's more, it is found that the CAP exposure time has a crucial effect on the plasma characteristics and cancer cell viability [22].
In order to help further understanding the cancer treatment using CAP jet, we try to study the simulation method of the cancer treatment using CAP jets. In this paper, we processed the experimental data from reference [23], then proposed the mathematic model using the processed data to study the relation between the cell viability, CAP jets exposure time and the treatment time. We

Results and Discussion
Here we have processed the experimental data from [23] in Tables 1 to 5      In order to investigate the cancer cell viability with time, we formulate a mathematical expression to cancer cell viability at different CAP exposure durations. Firstly, we got the average viability according to the tables 1 to 5, which is shown in Table 6. , , , , , a a a a a a are the coefficients to be determined for each CAP exposure time by the data in Table 6. We use the least squares method to process the discrepancy between the mathematical model and data in Table 6 to get function expression for each exposure time. That means the coefficients of 1 2 3 4 5 6 , , , , , a a a a a a were got by solving the following objective function.   We can conclude from figure 1 and figure 2 that the mathematical model can be used to predict the cancer cell viability under any CAP treatment conditions. We can just combine the formula (1) and formula (2) and calculate the coefficients using the least square method.
In order to have a better understanding of the response of cancer cell viability of U-87 MG for arbitrary CAP exposure duration, we also investigate the correlation of the cancer cell viability with the CAP exposure time and the treatment time.
Then, we use the Taylor expansion method to get the polynomial of the cancer cell viability related to the CAP exposure duration and the CAP treatment in the following expression (3).
( , ) = 0 + 1 + 2 ( ) 2 + 3 ⋅ + 4 7 0 = 2.229, 1 = 0.02233, 2 = 0.56, 3 = −0.0005, 4 = −0.002 Figure 3 shows the response of the cancer cell viability from generalized mathematical model (3) for arbitrary CAP treatment duration ∆t and time t. We can use formula (2)  (2) We also proposed the mathematic model using the Taylor expansion method using the average data to study the correlation between the cell viability, CAP exposure time and the treatment time. We can use it to predict the cancer cell viability for arbitrary CAP treatment duration ∆t and time t.
That is to say the cancer cell viability can be predict using the same method to under any CAP treatment conditions.

Material and Method
We proposed the mathematic model using the experimental data in Ref [23]. We select the experimental data of cancer cell line of U-87 MG under the CAP treatment. The CAP exposure duration is from 0 s to 180 s and the applied voltage is 3.16kV. We proposed the mathematical model to study the relation between the cell viability, CAP jet exposure time and the treatment 8 time.

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All data generated or analyzed during this study are included in this article.