Cisplatin is primarily considered to mediate cell death by targeting fast proliferating cells. It damages cellular DNA by cross linking the DNA of the cells to form DNA-cisplatin adducts [
23,
24]. The nucleotide excision repair (NER) pathway is activated in order to remove these adducts; however, upon failure of repair, cell apoptosis is triggered in mitotic cells and the dividing cells undergo cell death. Yet, recent studies have indicated that the cytotoxicity of cisplatin is not only limited to replicating cells. Cisplatin may elicit cell death independently of DNA damage via oxidative stress [
25,
26] at three primary locations [
27]. The first is the plasma membrane, where NOX induces reactive oxygen species (ROS) [
28] which in turn trigger Fas aggregation [
29] and influence the activity of membrane channels such as Ca
2+ channels [
28,
30]. Engagement and clustering of Fas receptor mediates caspase activation, while Ca
2+ influx changes electrolyte capacities of the cells [
28], which ultimately leads to cell apoptosis. Second, cytoplasm produces cellular superoxide formation through the interaction of cisplatin with nuclear DNA, thereby activating an important regulator of cell proliferation, differentiation and survival, the MAPK pathway [
27,
31]. The final and most important location is the mitochondrion, which generates free radical during oxidative phosphorylation and is one of the most important source of endogenous ROS [
27]. Treatment of cisplatin in NER deficient cells displayed similar mitochondrial ROS generation as NER proficient cells, demonstrating cisplatin-induced mitochondrial ROS production regardless of the ability of cells to repair cisplatin-induced nuclear DNA damage [
25]. At high doses of 10 μM or more, loss of mitochondrial membrane permeabilization, activation of BAK and caspases [
32] are triggered by superoxide formation induced by cisplatin [
25,
29], which is effectively blocked by antioxidants Manganese superoxide dismutase (MnSOD) and glutathione [
29,
33] and reactive oxygen species (ROS) scavengers [
32].
$$ \frac{\partial {\mathrm{C}}_{\mathrm{cis}}}{\mathrm{\partial t}}={\mathrm{D}}_{\mathrm{cis}}\left(\frac{\partial^2{\mathrm{C}}_{\mathrm{cis}}}{\partial {\mathrm{x}}^2}+\frac{\partial^2{\mathrm{C}}_{\mathrm{cis}}}{\partial {\mathrm{y}}^2}+\frac{\partial^2{\mathrm{C}}_{\mathrm{cis}}}{\partial {\mathrm{z}}^2}\right)-{\mathrm{K}}_{\mathrm{intra}}{\updelta}_{\mathrm{cell}}\ \left({\mathrm{C}}_{\mathrm{cis}}-{\mathrm{C}}_{\mathrm{intra}\mathrm{cis}}\right) $$
(7)
$$ \frac{\partial {\mathrm{C}}_{\mathrm{intra}\mathrm{cis}}}{\mathrm{\partial t}}={\mathrm{K}}_{\mathrm{intra}}{\updelta}_{\mathrm{cell}}\ \left({\mathrm{C}}_{\mathrm{cis}}-{\mathrm{C}}_{\mathrm{intra}\mathrm{cis}}\right)-{\mathrm{K}}_{\mathrm{bonding}}{\updelta}_{\mathrm{cell}}{\mathrm{C}}_{\mathrm{intra}\mathrm{cis}}-{\mathrm{K}}_{\mathrm{deg}}{\updelta}_{\mathrm{cell}}{\mathrm{C}}_{\mathrm{intra}\mathrm{cis}} $$
(8)
$$ \frac{\partial {\mathrm{C}}_{\mathrm{adduct}}}{\mathrm{\partial t}}={\mathrm{K}}_{\mathrm{bonding}}{\mathrm{C}}_{\mathrm{intracis}} $$
(9)
$$ {\updelta}_{\mathrm{c}\mathrm{ell}}=\left\{\begin{array}{c}0,\mathrm{where}\ {\mathrm{M}}_{\mathrm{c}}=0\\ {}1,\mathrm{where}\ {\mathrm{M}}_{\mathrm{c}}>0\end{array}\right\}\# $$
(10)
$$ {\mathrm{C}}_{\mathrm{adduct}}=\left\{\begin{array}{c}0,\mathrm{if}\ {\mathrm{P}}_{\mathrm{rep}}\left(\mathrm{t}\ \right)<\mathrm{R}\\ {}{\mathrm{C}}_{\mathrm{adduct}},\mathrm{if}\ {\mathrm{P}}_{\mathrm{rep}}\left(\mathrm{t}\right)\ge \mathrm{R}\end{array}\right\} $$
(11)
$$ {\displaystyle \begin{array}{c}{\mathrm{C}}_{\mathrm{l}}={\mathrm{C}}_{\mathrm{cismax}}\left(\frac{1}{\mathrm{A}+{\mathrm{Be}}^{-{\mathrm{t}}_{\mathrm{g}}}}\right)\\ {}\ \mathrm{here},{\mathrm{t}}_{\mathrm{g}}=\left({\mathrm{t}}_{\mathrm{m}}-{\mathrm{t}}_{\mathrm{cg}}\right)\ \end{array}} $$
(12)
$$ {\mathrm{M}}_{\mathrm{c}}=0,\mathrm{when}\ {\mathrm{C}}_{\mathrm{adduct}}>{\mathrm{C}}_{\mathrm{l}} $$
(13)
The effects of cisplatin on the cells are numerically modelled by assuming that cancer cells experiencing oxidative stress suffer DNA damage upon treatment with cisplatin. DNA damage builds-up during the course of the treatment. Therefore, ROS formation in the model correlates with the quantity of cisplatin uptake by the cells. The drug uptake is modelled by eqs.
7–
10, where
A and
B are constants. To model the uptake of cisplatin more accurately, cisplatin concentration is split into two parts, cisplatin concentration outside the cell or extracellular concentration
(Ccis), with diffusion coefficient (
Dcis) and concentration inside the cell or intracellular concentration (
Cintracis) with passive diffusion rate constant (
Kintra). The intracellular cisplatin degrades based on the degradation rate constant (
Kdeg). The rate of formation of DNA adducts (
Cadduct) is assumed to be directly proportional to the intracellular concentration with a rate constant (
Kbonding). The cells attempt to remove excess ROS via their endogenous antioxidant defence mechanism in-vivo and in-vitro. In accordance with these experimental observations, in the model, the cells eliminate the adducts formed based on the probability ‘
R’ and
Prep(
t) as shown in eq.
11. The cells in the model are assumed to repair themselves with equal probability ‘
R’. Cells exposed to cisplatin respond differently based on their exposure times to the drug. Therefore, an apparent lethal concentration
Cl is calculated based on eq.
12, which is dependent on the cell’s cumulative generational age
tcg, maximum generational age for mutation
tm, and the age-independent maximum adduct concentration
Ccismax as shown in eq.
12. Cumulative generational age is the total existing time of the cell’s generation. The cumulative age is divided equally between the daughter cells at the time of division. In the model, we assume no mutation of the cells and therefore do not consider the effects of acquired drug resistance to cell lysis. Therefore, we include the maximum generational age parameter
tm, which imposes maximum probability of cell lysis for a long-lasting cell generation capable of mutating. This sets an artificial time boundary after which none of cells are allowed to exist in the simulation and accumulate mutations. If the adduct concentration
Cadduct is larger than the permissible lethal adduct concentration
Cl, after all possible cell repair processes, the cells undergoes apoptosis. However, if the cell is able to overcome cisplatin cytotoxicity, in-vivo or in-vitro, with its NER pathway, and reduce ROS through its antioxidant defence, the cell mass divides with the oxidative stress level lesser than the lethal threshold. This phenomenon is modelled in such a way that both the daughter cell and the parent cell do not inherit any DNA damage, that is,
Cadduct and
Ccis are reset.