Tamara Rubio González

Distributions of the potential and electric field of an electrode elliptic array used in tumor electrotherapy: Analytical and numerical solutions

To estimate the potential and electric field generated by any electrode array is very useful in effective tumor destruction. At present, an electrode array that takes into account the ellipsoidal geometry of the solid tumors has not been proposed. We present both analytical and numerical solutions for the potential and electric field in a solid tumor established by an electrode array with elliptic shape which may be used in vitro, in vivo and in clinical studies for cancer treatment with electrotherapy. These analytical and numerical solutions are obtained using multipole expansion and the finite difference method. Distributions of potential and electric field magnitudes are computed in function of the eccentricity of an elliptical array and compared with those obtained with a circular array of electrode. Maximum difference and Root Means Square Error are used to compare the distributions of the potential and electric field in leading-order and first-order correction and between the analytical and numerical solutions. The results show a good agreement between these distributions in both orders and the analytical and numerical solutions. It was concluded that the mathematical approach presented in this study is a tool for a rapid design of electrode elliptical arrays in order to induce the maximum destruction of the tumor. Moreover, it is shown that, for all values of eccentricity, there is a good correspondence between the distributions of the potential and the electric field for leading-order and first-order correction and for both the analytical and numerical solutions.

Modified Gompertz Equation for Electrotherapy Murine Tumor Growth Kinetics: Predictions and New Hypotheses

BackgroundElectrotherapy 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.MethodsThe 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 tumors natural history before treatment. Different graphical strategies are used in order to reveal new information in the complete kinetics of this tumor type.ResultsThe 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.ConclusionThe 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.

Is cancer a pure growth curve or does it follow a kinetics of dynamical structural transformation

BackgroundUnperturbed tumor growth kinetics is one of the more studied cancer topics; however, it is poorly understood. Mathematical modeling is a useful tool to elucidate new mechanisms involved in tumor growth kinetics, which can be relevant to understand cancer genesis and select the most suitable treatment.MethodsThe classical Kolmogorov-Johnson-Mehl-Avrami as well as the modified Kolmogorov-Johnson-Mehl-Avrami models to describe unperturbed fibrosarcoma Sa-37 tumor growth are used and compared with the Gompertz modified and Logistic models. Viable tumor cells (1×105) are inoculated to 28 BALB/c male mice.ResultsModified Gompertz, Logistic, Kolmogorov-Johnson-Mehl-Avrami classical and modified Kolmogorov-Johnson-Mehl-Avrami models fit well to the experimental data and agree with one another. A jump in the time behaviors of the instantaneous slopes of classical and modified Kolmogorov-Johnson-Mehl-Avrami models and high values of these instantaneous slopes at very early stages of tumor growth kinetics are observed.ConclusionsThe modified Kolmogorov-Johnson-Mehl-Avrami equation can be used to describe unperturbed fibrosarcoma Sa-37 tumor growth. It reveals that diffusion-controlled nucleation/growth and impingement mechanisms are involved in tumor growth kinetics. On the other hand, tumor development kinetics reveals dynamical structural transformations rather than a pure growth curve. Tumor fractal property prevails during entire TGK.

Original article: Influence of electrode array parameters used in electrotherapy on tumor growth kinetics: A mathematical simulation

Evaluation of the distance between the electrodes, voltage applied to them, and number of electrodes in tumor growth kinetics is very useful for effective tumor destruction when electrotherapy is used. However, a study of this type has not yet been proposed. The aim of this paper is to simulate the influence of such parameters and the point-point electrode configuration on the tumor growth kinetics through a Modified Gompertz Equation. The results show a good agreement between the simulations performed in this study and the experimental results reported by our group and other authors. A critical distance between electrodes and a threshold ratio between the applied electric field and that distributed in the tumor are revealed, for which higher electrotherapy antitumor effectiveness is reached. In conclusion, electrotherapy antitumor effectiveness not only depends on the distance between the electrodes, voltage applied to them, and number of electrodes, but also on the ratio between the applied electric field and that distributed in the tumor. In addition, the results of these simulations may be used to help physicians choose the most appropriate treatment for patients with malignant solid tumors, as we have implemented in a current clinical trial.

Dose-response study for the highly aggressive and metastatic primary F3II mammary carcinoma under direct current: Direct Current on Highly Aggressive Primary Tumors

Electrochemical treatment has been suggested as an effective alternative to local cancer therapy. Nevertheless, its effectiveness decreases when highly aggressive primary tumors are treated. The aim of this research was to understand the growth kinetics of the highly aggressive and metastatic primary F3II tumor growing in male and female BALB/c/Cenp mice under electrochemical treatment. Different amounts of electric charge (6, 9, and 18 C) were used. Two electrodes were inserted into the base, perpendicular to the tumors long axis, keeping about 1 cm distance between them. Results have shown that the F3II tumor is highly sensitive to direct current. The overall effectiveness (complete response + partial response) of this physical agent was ≥75.0% and observed in 59.3% (16/27) of treated F3II tumors. Complete remission of treated tumors was observed in 22.2% (6/27). An unexpected result was the death of 11 direct current-treated animals (eight females and three males). It is concluded that direct current may be addressed to significantly affect highly aggressive and metastatic primary tumor growth kinetics, including the tumor complete response. Bioelectromagnetics. 39:460-475, 2018.