Simulations of surface charge density changes during the untreated solid tumour growth.

Understanding untreated tumour growth kinetics and its intrinsic behaviour is interesting and intriguing. The aim of this study is to propose an approximate analytical expression that allows us to simulate changes in surface charge density at the cancer-surrounding healthy tissue interface during the untreated solid tumour growth. For this, the Gompertz and Poisson equations are used. Simulations reveal that the unperturbed solid tumour growth is closely related to changes in the surface charge density over time between the tumour and the surrounding healthy tissue. Furthermore, the unperturbed solid tumour growth is governed by temporal changes in this surface charge density. It is concluded that results corroborate the correspondence between the electrical and physiological parameters in the untreated cancer, which may have an essential role in its growth, progression, metastasis and protection against immune system attack and anti-cancer therapies. In addition, the knowledge of surface charge density changes at the cancer-surrounding healthy tissue interface may be relevant when redesigning the molecules in chemotherapy and immunotherapy taking into account their polarities. This can also be true in the design of completely novel therapies.

Efficacy of direct current generated by multiple-electrode arrays on F3II mammary carcinoma: experiment and mathematical modeling.

BACKGROUND: The modified Gompertz equation has been proposed to fit experimental data for direct current treated tumors when multiple-straight needle electrodes are individually inserted into the base perpendicular to the tumor long axis. The aim of this work is to evaluate the efficacy of direct current generated by multiple-electrode arrays on F3II mammary carcinoma that grow in the male and female BALB/c/Cenp mice, when multiple-straight needle electrodes and multiple-pairs of electrodes are inserted in the tumor. METHODS: A longitudinal and retrospective preclinical study was carried out. Male and female BALB/c/Cenp mice, the modified Gompertz equation, intensities (2, 6 and 10 mA) and exposure times (10 and 20 min) of direct current, and three geometries of multiple-electrodes (one formed by collinear electrodes and two by pair-electrodes) were used. Tumor volume and mice weight were measured. In addition, the mean tumor doubling time, tumor regression percentage, tumor growth delay, direct current overall effectiveness and mice survival were calculated. RESULTS: The greatest growth retardation, mean doubling time, regression percentage and growth delay of the primary F3II mammary carcinoma in male and female mice were observed when the geometry of multiple-pairs of electrodes was arranged in the tumor at 45, 135, 225 and 325o and the longest exposure time. In addition, highest direct current overall effectiveness (above 66%) was observed for this EChT scheme. CONCLUSIONS: It is concluded that electrochemical therapy may be potentially addressed to highly aggressive and metastic primary F3II murine mammary carcinoma and the modified Gompertz equation may be used to fit data of this direct current treated carcinoma. Additionally, electrochemical therapy effectiveness depends on the exposure time, geometry of multiple-electrodes and ratio between the direct current intensity applied and the polarization current induced in the tumor.

New formulation of the Gompertz equation to describe the kinetics of untreated tumors.

BACKGROUND: Different equations have been used to describe and understand the growth kinetics of undisturbed malignant solid tumors. The aim of this paper is to propose a new formulation of the Gompertz equation in terms of different parameters of a malignant tumor: the intrinsic growth rate, the deceleration factor, the apoptosis rate, the number of cells corresponding to the tumor latency time, and the fractal dimensions of the tumor and its contour. METHODS: Furthermore, different formulations of the Gompertz equation are used to fit experimental data of the Ehrlich and fibrosarcoma Sa-37 tumors that grow in male BALB/c/Cenp mice. The parameters of each equation are obtained from these fittings. RESULTS: The new formulation of the Gompertz equation reveals that the initial number of cancerous cells in the conventional Gompertz equation is not a constant but a variable that depends nonlinearly on time and the tumor deceleration factor. In turn, this deceleration factor depends on the apoptosis rate of tumor cells and the fractal dimensions of the tumor and its irregular contour. CONCLUSIONS: It is concluded that this new formulation has two parameters that are directly estimated from the experiment, describes well the growth kinetics of unperturbed Ehrlich and fibrosarcoma Sa-37 tumors, and confirms the fractal origin of the Gompertz formulation and the fractal property of tumors.

Tissue Damage, Temperature, and pH Induced by Different Electrode Arrays on Potato Pieces (Solanum tuberosum L.).

One of the most challenging problems of electrochemical therapy is the design and selection of suitable electrode array for cancer. The aim is to determine how two-dimensional spatial patterns of tissue damage, temperature, and pH induced in pieces of potato (Solanum tuberosum L., var. Mondial) depend on electrode array with circular, elliptical, parabolic, and hyperbolic shape. The results show the similarity between the shapes of spatial patterns of tissue damage and electric field intensity, which, like temperature and pH take the same shape of electrode array. The adequate selection of suitable electrodes array requires an integrated analysis that involves, in a unified way, relevant information about the electrochemical process, which is essential to perform more efficiently way the therapeutic planning and the personalized therapy for patients with a cancerous tumor.

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

BACKGROUND: Unperturbed 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. METHODS: The 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. RESULTS: Modified 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. CONCLUSIONS: The 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.

Antitumor effects of electrochemical treatment.

Electrochemical treatment is an alternative modality for tumor treatment based on the application of a low intensity direct electric current to the tumor tissue through two or more platinum electrodes placed within the tumor zone or in the surrounding areas. This treatment is noted for its great effectiveness, minimal invasiveness and local effect. Several studies have been conducted worldwide to evaluate the antitumoral effect of this therapy. In all these studies a variety of biochemical and physiological responses of tumors to the applied treatment have been obtained. By this reason, researchers have suggested various mechanisms to explain how direct electric current destroys tumor cells. Although, it is generally accepted this treatment induces electrolysis, electroosmosis and electroporation in tumoral tissues. However, action mechanism of this alternative modality on the tumor tissue is not well understood. Although the principle of Electrochemical treatment is simple, a standardized method is not yet available. The mechanism by which Electrochemical treatment affects tumor growth and survival may represent more complex process. The present work analyzes the latest and most important research done on the electrochemical treatment of tumors. We conclude with our point of view about the destruction mechanism features of this alternative therapy. Also, we suggest some mechanisms and strategies from the thermodynamic point of view for this therapy. In the area of Electrochemical treatment of cancer this tool has been exploited very little and much work remains to be done. Electrochemical treatment constitutes a good therapeutic option for patients that have failed the conventional oncology methods.

3D Stationary electric current density in a spherical tumor treated with low direct current: an analytical solution.

Electrotherapy with direct current delivered through implanted electrodes is used for local control of solid tumors in both preclinical and clinical studies. The aim of this research is to develop a solution method for obtaining a three-dimensional analytical expression for potential and electric current density as functions of direct electric current intensity, differences in conductivities between the tumor and the surrounding healthy tissue, and length, number and polarity of electrodes. The influence of these parameters on electric current density in both media is analyzed. The results show that the electric current density in the tumor is higher than that in the surrounding healthy tissue for any value of these parameters. The conclusion is that the solution method presented in this study is of practical interest because it provides, in a few minutes, a convenient way to visualize in 3D the electric current densities generated by a radial electrode array by means of the adequate selection of direct current intensity, length, number, and polarity of electrodes, and the difference in conductivity between the solid tumor and its surrounding healthy tissue.