Modified Gompertz equation for electrotherapy murine tumor growth kinetics: predictions and new hypotheses

Abstract

Background: Electrotherapy effectiveness at different doses has been demonstrated in preclinical and clinical studies; however, several aspects that occur in the tumor growth kinetics before and after treatment have not yet been revealed. Mathematical modeling is a useful instrument that can reveal some of these aspects. The aim of this paper is to describe the complete growth kinetics of unperturbed and perturbed tumors through use of the modified Gompertz equation in order to generate useful insight into the mechanisms that underpin this devastating disease.

Methods: The complete tumor growth kinetics for control and treated groups are obtained by interpolation and extrapolation methods with different time steps, using experimental data of fibrosarcoma Sa-37. In the modified Gompertz equation, a delay time is introduced to describe the tumor's natural history before treatment. Different graphical strategies are used in order to reveal new information in the complete kinetics of this tumor type.

Results: The first stage of complete tumor growth kinetics is highly non linear. The model, at this stage, shows different aspects that agree with those reported theoretically and experimentally. Tumor reversibility and the proportionality between regions before and after electrotherapy are demonstrated. In tumors that reach partial remission, two antagonistic post-treatment processes are induced, whereas in complete remission, two unknown antitumor mechanisms are induced.

Conclusion: The modified Gompertz equation is likely to lead to insights within cancer research. Such insights hold promise for increasing our understanding of tumors as self-organizing systems and, the possible existence of phase transitions in tumor growth kinetics, which, in turn, may have significant impacts both on cancer research and on clinical practice.

Figure 1

Time dependence of TV: Unperturbed fibrosarcoma Sa-37 TGK (CG) for the parameters i = 0 mA, α = 0.513 days^-1, β = 0.262 days^-1, V_o = 0.5 cm³ and a time step of Δt = 1/3 days.

Figure 2

TV dependence of the FDTV: Unperturbed fibrosarcoma Sa-37 TGK (CG) for the parameters i = 0 mA, α = 0.513 days^-1, β = 0.262 days^-1, V_o = 0.5 cm³ and a time step of Δt = 1/3 days.

Figure 3

Time dependence of TV: DEC-perturbed fibrosarcoma Sa-37 TGK (TG2) for the parameters i = 11.7 mA, α = 1.584 days^-1, β = 0.076 days^-1, γ = 0.107 days^-1, i_o = 7.431 mA, and V_o = 0.5 cm³ and a time step of Δt = 1/3 days.

Figure 4

TV dependence of the FDTV: DEC-perturbed fibrosarcoma Sa-37 TGK (TG2) for the parameters i = 11.7 mA, α = 1.584 days^-1, β = 0.076 days^-1, γ = 0.107 days^-1, i_o = 7.431 mA, and V_o = 0.5 cm³ and a time step of Δt = 1/3 days.

Figure 5

Time dependence of TV: DEC-perturbed fibrosarcoma Sa-37 TGK (TG3) for the parameters i = 14.8 mA, α = 0.006 days^-1, β = 0.207 days^-1, γ = 0.189 days^-1, i_o = 1.080 mA, and V_o = 0.5 cm³ and a time step of Δt = 1/3 days.

Figure 6

TV dependence of the FDTV: DEC-perturbed fibrosarcoma Sa-37 TGK (TG3) for the parameters i = 14.8 mA, α = 0.006 days^-1, β = 0.207 days^-1, γ = 0.189 days^-1, i_o = 1.080 mA, and V_o = 0.5 cm³ and a time step of Δt = 1/3 days.

Figure 7

Time consecutive dependence of the TV plot (V*(t) vs. V*(t-Δt) plot) for Δt = 1/3 and 1/24 days.

Figure 8

Analysis conducted by separating REG-I and REG-II in a plot of module of FDTV versus TV.

Figure 9

Schematic representation of TGK.

Figure 10

Representation equivalent of TGK from a biophysical point of view.