The Quest to Quantify Selective and Synergistic Effects of Plasma for Cancer Treatment: Insights from Mathematical Modeling

Abstract

Cold atmospheric plasma (CAP) and plasma-treated liquids (PTLs) have recently become a promising option for cancer treatment, but the underlying mechanisms of the anti-cancer effect are still to a large extent unknown. Although hydrogen peroxide (H2O2) has been recognized as the major anti-cancer agent of PTL and may enable selectivity in a certain concentration regime, the co-existence of nitrite can create a synergistic effect. We develop a mathematical model to describe the key species and features of the cellular response toward PTL. From the numerical solutions, we define a number of dependent variables, which represent feasible measures to quantify cell susceptibility in terms of the H2O2 membrane diffusion rate constant and the intracellular catalase concentration. For each of these dependent variables, we investigate the regimes of selective versus non-selective, and of synergistic versus non-synergistic effect to evaluate their potential role as a measure of cell susceptibility. Our results suggest that the maximal intracellular H2O2 concentration, which in the selective regime is almost four times greater for the most susceptible cells compared to the most resistant cells, could be used to quantify the cell susceptibility toward exogenous H2O2. We believe our theoretical approach brings novelty to the field of plasma oncology, and more broadly, to the field of redox biology, by proposing new ways to quantify the selective and synergistic anti-cancer effect of PTL in terms of inherent cell features.

Keywords: cold atmospheric plasma; hydrogen peroxide; mathematical modeling; reaction network; selective cancer treatment.

Figure A1

The dependent variable $c_{4,\max }$ (i.e., the temporal maximum of $\left[ONO{O}^{-}\right]$ in the IC) as a function of $k_{D,1}$ and ${[CATF{e}^{III}]}_0$ when ${\left[H_2{O}_2\right]}_0^{EC}=1 \mathsf{\mu }\mathrm{M}$. ${\left[N{O}_2^{-}\right]}_0^{EC}=0 \mathrm{M}$ (a) and ${\left[N{O}_2^{-}\right]}_0^{EC}=1 \mathrm{mM}$ (b).

Figure A2

The dependent variable $c_{4,\max }$ (i.e., the temporal maximum of $\left[ONO{O}^{-}\right]$ in the IC) as a function of $k_{D,1}$ and ${[CATF{e}^{III}]}_0$ when ${\left[H_2{O}_2\right]}_0^{EC}=1 \mathrm{mM}$. ${\left[N{O}_2^{-}\right]}_0^{EC}=0 \mathrm{M}$ (a) and ${\left[N{O}_2^{-}\right]}_0^{EC}=1 \mathrm{mM}$ (b).

Figure A3

The dependent variable $\tau$ (i.e., the system response time of $\left[H_2{O}_2\right]$ in the IC) as a function of $k_{D,1}$ and ${[CATF{e}^{III}]}_0$ when ${\left[H_2{O}_2\right]}_0^{EC}=1 \mathsf{\mu }\mathrm{M}$. ${\left[N{O}_2^{-}\right]}_0^{EC}=0 \mathrm{M}$ (a) and ${\left[N{O}_2^{-}\right]}_0^{EC}=1 \mathrm{mM}$ (b).

Figure A4

The dependent variable $\tau$ (i.e., the system response time of $\left[H_2{O}_2\right]$ in the IC) as a function of $k_{D,1}$ and ${[CATF{e}^{III}]}_0$ when ${\left[H_2{O}_2\right]}_0^{EC}=1 \mathrm{mM}$. ${\left[N{O}_2^{-}\right]}_0^{EC}=0 \mathrm{M}$ (a) and ${\left[N{O}_2^{-}\right]}_0^{EC}=1 \mathrm{mM}$ (b).

Figure A5

The dependent variable $l_1$ (i.e., the load of $H_2{O}_2$ in the IC) as a function of $k_{D,1}$ and ${[CATF{e}^{III}]}_0$ when ${\left[H_2{O}_2\right]}_0^{EC}=1 \mathsf{\mu }\mathrm{M}$. ${\left[N{O}_2^{-}\right]}_0^{EC}=0 \mathrm{M}$ (a) and ${\left[N{O}_2^{-}\right]}_0^{EC}=1 \mathrm{mM}$ (b).

Figure A6

The dependent variable $l_1$ (i.e., the load of $H_2{O}_2$ in the IC) as a function of $k_{D,1}$ and ${[CATF{e}^{III}]}_0$ when ${\left[H_2{O}_2\right]}_0^{EC}=1 \mathrm{mM}$. ${\left[N{O}_2^{-}\right]}_0^{EC}=0 \mathrm{M}$ (a) and ${\left[N{O}_2^{-}\right]}_0^{EC}=1 \mathrm{mM}$ (b).

Figure A7

The dependent variable $l_{1,BS}$ (i.e., the load over the baseline of $H_2{O}_2$ in the IC) as a function of $k_{D,1}$ and ${[CATF{e}^{III}]}_0$ when ${\left[H_2{O}_2\right]}_0^{EC}=1 \mathsf{\mu }\mathrm{M}$. ${\left[N{O}_2^{-}\right]}_0^{EC}=0 \mathrm{M}$ (a) and ${\left[N{O}_2^{-}\right]}_0^{EC}=1 \mathrm{mM}$ (b).

Figure A8

The dependent variable $l_{1,BS}$ (i.e., the load over the baseline of $H_2{O}_2$ in the IC) as a function of $k_{D,1}$ and ${[CATF{e}^{III}]}_0$ when ${\left[H_2{O}_2\right]}_0^{EC}=1 \mathrm{mM}$. ${\left[N{O}_2^{-}\right]}_0^{EC}=0 \mathrm{M}$ (a) and ${\left[N{O}_2^{-}\right]}_0^{EC}=1 \mathrm{mM}$ (b).

Figure A9

The dependent variable $l_4$ (i.e., the load of $ONO{O}^{-}$ in the IC) as a function of $k_{D,1}$ and ${[CATF{e}^{III}]}_0$ when ${\left[H_2{O}_2\right]}_0^{EC}=1 \mathsf{\mu }\mathrm{M}$. ${\left[N{O}_2^{-}\right]}_0^{EC}=0 \mathrm{M}$ (a) and ${\left[N{O}_2^{-}\right]}_0^{EC}=1 \mathrm{mM}$ (b).

Figure A10

The dependent variable $l_4$ (i.e., the load of $ONO{O}^{-}$ in the IC) as a function of $k_{D,1}$ and ${[CATF{e}^{III}]}_0$ when ${\left[H_2{O}_2\right]}_0^{EC}=1 \mathrm{mM}$. ${\left[N{O}_2^{-}\right]}_0^{EC}=0 \mathrm{M}$ (a) and ${\left[N{O}_2^{-}\right]}_0^{EC}=1 \mathrm{mM}$ (b).

Figure A11

The dependent variable $\overline{s}$ (i.e., the inverse of the average rate of $H_2{O}_2$ consumption in the EC) as a function of $k_{D,1}$ and ${[CATF{e}^{III}]}_0$ when ${\left[H_2{O}_2\right]}_0^{EC}=1 \mathsf{\mu }\mathrm{M}$. ${\left[N{O}_2^{-}\right]}_0^{EC}=0 \mathrm{M}$ (a) and ${\left[N{O}_2^{-}\right]}_0^{EC}=1 \mathrm{mM}$ (b).

Figure A12

The dependent variable $\overline{s}$ (i.e., the inverse of the average rate of $H_2{O}_2$ consumption in the EC) as a function of $k_{D,1}$ and ${[CATF{e}^{III}]}_0$ when ${\left[H_2{O}_2\right]}_0^{EC}=1\mathrm{mM}$. ${\left[N{O}_2^{-}\right]}_0^{EC}=0 \mathrm{M}$ (a) and ${\left[N{O}_2^{-}\right]}_0^{EC}=1 \mathrm{mM}$ (b).

Figure A13

The dependent variable $s_{\max }$ (i.e., the inverse of the maximal rate of $H_2{O}_2$ consumption in the EC) as a function of $k_{D,1}$ and ${[CATF{e}^{III}]}_0$ when ${\left[H_2{O}_2\right]}_0^{EC}=1 \mathsf{\mu }\mathrm{M}$. ${\left[N{O}_2^{-}\right]}_0^{EC}=0 \mathrm{M}$ (a) and ${\left[N{O}_2^{-}\right]}_0^{EC}=1 \mathrm{mM}$ (b).

Figure A14

The dependent variable $s_{\max }$ (i.e., the inverse of the maximal rate of $H_2{O}_2$ consumption in the EC) as a function of $k_{D,1}$ and ${[CATF{e}^{III}]}_0$ when ${\left[H_2{O}_2\right]}_0^{EC}=1 \mathrm{mM}$. ${\left[N{O}_2^{-}\right]}_0^{EC}=0 \mathrm{M}$ (a) and ${\left[N{O}_2^{-}\right]}_0^{EC}=1 \mathrm{mM}$ (b).

Figure A15

Illustration of a linear concentration gradient over a membrane.

Figure 1

Illustration of the system representing a cell exposed to PTL.

Figure 2

The dependent variable $c_{1,\max }$ (i.e., the temporal maximum of $\left[H_2{O}_2\right]$ in the IC) as a function of $k_{D,1}$ and ${[CATF{e}^{III}]}_0$ when ${\left[H_2{O}_2\right]}_0^{EC}=1 \mathsf{\mu }\mathrm{M}$. ${\left[N{O}_2^{-}\right]}_0^{EC}=0 \mathrm{M}$ (a) and ${\left[N{O}_2^{-}\right]}_0^{EC}=1 \mathrm{mM}$ (b).

Figure 3

The dependent variable $c_{1,\max }$ (i.e., the temporal maximum of $\left[H_2{O}_2\right]$ in the IC) as a function of $k_{D,1}$ and ${[CATF{e}^{III}]}_0$ when ${\left[H_2{O}_2\right]}_0^{EC}=1 \mathrm{mM}$. ${\left[N{O}_2^{-}\right]}_0^{EC}=0 \mathrm{M}$ (a) and ${\left[N{O}_2^{-}\right]}_0^{EC}=1 \mathrm{mM}$ (b).

Figure 4

The dependent variable $c_{1,\max }$ (i.e., the temporal maximum of $\left[H_2{O}_2\right]$ in the IC) as a function of $k_{D,1}$ and $\log \left({[CATF{e}^{III}]}_0\right)$ when ${\left[H_2{O}_2\right]}_0^{EC}=1 \mathsf{\mu }\mathrm{M}$ and ${\left[N{O}_2^{-}\right]}_0^{EC}=0 \mathrm{M}$.