Modelling of macrophages interactions in breast cancer by partial differential equations

Tthe signalling interaction between tumor cells and macrophages will form spontaneous aggregation that causes tumor spreading. tumor cells and macrophages interchange their respective signals which results a paracrine and autocrine signalling loop. this interaction process can be represented by mathematical model in the form of chemotaxis and reaction diffusion equations. the existing models that consider paracrine signalling loop alone or with the inclusion of paracrine and autocrine signalling loop had been developed by assuming linear signals production. however, this assumption does not give a better representation on the signal dynamics where it is supposed to be in nonlinear form that saturate with increasing cell densities. therefore, in this research, two existing interaction models are improved by considering the nonlinear form of signals production. besides, another new interaction model is also developed based on the facts that tumor cells release enzyme during the signaling interaction to penetrate the surrounding tissues. the stability analysis is conducted on three separated models to investigate the condition for spontaneous aggregation. each of these conditions then are validated using numerical simulations. stability analysis shows that for all models, the formation of aggregation could be determined by the parameter that represents the secretion and degradation rates of signals together with chemotaxis rates towards signals. however, the inclusion of autocrine signalling loop in the second model increase the possibility of the aggregation. while in the third model, an additional parameter that represents the secretion and degradation rates of enzyme as well as chemotaxis rates towards them could also determine the formation of the aggregation. by numerical simulations, the results are in agreement with the stability analysis obtained for each of the interaction models. besides, cell clusters that result from the aggregation will be merged to the other cells cluster due to the “effective attraction” between them. reducing the production rates of signal or chemotaxis rates towards signals or increasing degradation rates of signal is required to prevent aggregation. the same changes towards enzymes will give the same result on preventing the aggregation. these valuable suggestions are crucial for medical experts during treatments

A new modern scheme for solving fractal–fractional differential equations based on deep feedforward neural network with multiple hidden layer

The recent development of knowledge in fractional calculus introduced an advanced superior operator known as fractal–fractional derivative (FFD). This operator combines memory effect and self-similar property that give better accurate representation of real world problems through fractal–fractional differential equations (FFDEs). However, the existence of fresh and modern numerical technique on solving FFDEs is still scarce. Originally invented for machine learning technique, artificial neural network (ANN) is cutting-edge scheme that have shown promising result in solving the fractional differential equations (FDEs). Thus, this research aims to extend the application of ANN to solve FFDE with power law kernel in Caputo sense (FFDEPC) by develop a vectorized algorithm based on deep feedforward neural network that consists of multiple hidden layer (DFNN-2H) with Adam optimization. During the initial stage of the method development, the basic framework on solving FFDEs is designed. To minimize the burden of computational time, the vectorized algorithm is constructed at the next stage for method to be performed efficiently. Several example have been tested to demonstrate the applicability and efficiency of the method. Comparison on exact solutions and some previous published method indicate that the proposed scheme have give good accuracy and low computational time