Fractional Online Learning Rate: Influence of Psychological Factors on Learning Acquisition.

The quantification of learning acquisition in a blended and online course is still slightly explored from the complex systems lens. The fractional online learning rate (fOLR) using fractional integrals is introduced. The notion of fOLR is based on the nonlinearity of the individual students learning pathway network, built from Learning Management System log files. Several learning pathway networks from students that pass or fail the course were constructed. The Akaike information criterion shows that the minimum number of boxes to cover these networks follow a power-law model. Further analysis shows that the fOLR model and its parameters were significantly compared with the online learning rate model. Thus, the fOLR was computing power and delayed power models, inspired by the "law of practice." The results show that the fractional definition is a better model and has a nonlinear relationship with the overall grade. Also, engagement and disengagement mould the fOLR curve. It means that the student's performance is affected by the engagement, and it is necessary that they are encouraged to pay more effort and attention to the learning activities, and those activities need to be designed to be fun and pleasant to improve the learning achievements.

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.

Analytical and numerical solutions of the potential and electric field generated by different electrode arrays in a tumor tissue under electrotherapy.

BACKGROUND: Electrotherapy is a relatively well established and efficient method of tumor treatment. In this paper we focus on analytical and numerical calculations of the potential and electric field distributions inside a tumor tissue in a two-dimensional model (2D-model) generated by means of electrode arrays with shapes of different conic sections (ellipse, parabola and hyperbola). METHODS: Analytical calculations of the potential and electric field distributions based on 2D-models for different electrode arrays are performed by solving the Laplace equation, meanwhile the numerical solution is solved by means of finite element method in two dimensions. RESULTS: Both analytical and numerical solutions reveal significant differences between the electric field distributions generated by electrode arrays with shapes of circle and different conic sections (elliptic, parabolic and hyperbolic). Electrode arrays with circular, elliptical and hyperbolic shapes have the advantage of concentrating the electric field lines in the tumor. CONCLUSION: The mathematical approach presented in this study provides a useful tool for the design of electrode arrays with different shapes of conic sections by means of the use of the unifying principle. At the same time, we verify the good correspondence between the analytical and numerical solutions for the potential and electric field distributions generated by the electrode array with different conic sections.