Modeling the Effects of Chemotherapy and Immunotherapy on Tumor Growth

2021 ◽  
Vol 17 (12) ◽  
pp. 2505-2518
Author(s):  
Sara El Haout ◽  
Maymunah Fatani ◽  
Nadia Abu Farha ◽  
Nour AlSawaftah ◽  
Maruf Mortula ◽  
...  
Keyword(s):  
Mathematical Modeling ◽  
Mathematical Model ◽  
Cd8 T Cells ◽  
The Other ◽  
Kutta Method ◽  
Cancer Treatments ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
Combined Intervention ◽  
Combined Interventions

Mathematical modeling has been used to simulate the interaction of chemotherapy and immunotherapy drugs intervention with the dynamics of tumor cells growth. This work studies the interaction of cells in the immune system, such as the natural killer, dendritic, and cytotoxic CD8+ T cells, with chemotherapy. Four different cases were considered in the simulation: no drug intervention, independent interventions (either chemotherapy or immunotherapy), and combined interventions of chemotherapy and immunotherapy. The system of ordinary differential equations was initially solved using the Runge-Kutta method and compared with two additional methods: the Explicit Euler and Heun’s methods. Results showed that the combined intervention is more effective compared to the other cases. In addition, when compared with Runge-Kutta, the Heun’s method presented a better accuracy than the Explicit Euler technique. The proposed mathematical model can be used as a tool to improve cancer treatments and targeted therapy.

A modified Runge-Kutta method

SIMULATION ◽  
1968 ◽  
Vol 10 (5) ◽  
pp. 221-223 ◽  
Author(s):  
A.S. Chai
Keyword(s):  
Error Estimate ◽  
Linear Combination ◽  
Experimental Results ◽  
The Other ◽  
New Method ◽  
Kutta Method ◽  
Starting Values ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
A Minor

It is possible to replace k2 in a 4th-order Runge-Kutta for mula (also Nth-order 3 ≤ N ≤ 5) by a linear combination of k1 and the ki's in the last step, using the same procedure for computing the other ki's and y as in the standard R-K method. The advantages of the new method are: It re quires one less derivative evaluation, provides an error estimate at each step, gives more accurate results, and needs a minor change to switch to the RK to obtain the starting values. Experimental results are shown in verification of the for mula.

Nanofiber formation in the presence of an external magnetic field in electrospinning

2015 ◽  
Vol 35 (6) ◽  
pp. 587-596 ◽  
Author(s):  
Saeide S. Badieyan ◽  
Mohsen Janmaleki
Keyword(s):  
Magnetic Field ◽  
Mathematical Model ◽  
External Magnetic Field ◽  
Charged Particles ◽  
Electrospinning Process ◽  
Kutta Method ◽  
Rich Variety ◽  
Runge Kutta Method ◽  
Runge Kutta

Abstract Electrospinning is an efficient, versatile, and straightforward technique involving the fabrication of very thin fibers from a rich variety of materials. Despite several promising applications, the remaining problem with electrospinning is the unpredictable deposition of the nanofibers. In this study, a mathematical model for a novel magnetic electrospinning process was established on the basis of a set of equations. Then, the model was utilized to analyze the behavior of the created polymer jet numerically using the Runge-Kutta method. The jet was assumed to consist of a number of discrete charged particles connected by viscoelastic segments. The results showed that exerting an appropriate magnetic field (MF) could significantly decrease the radius and the instability of the whipping circles. After fixing the instability as far as possible, it was demonstrated that a properly applied perpendicular MF could largely adjust the target of the polymer jet on the collector.

scholarly journals Some Numerical Methods and Comparisons for Solving Mathematical Model of Surface Decontamination by Disinfectant Solution

MATEMATIKA ◽  
2018 ◽  
Vol 34 (2) ◽  
pp. 271-291
Author(s):  
Chai Jin Sian ◽  
Yeak Su Hoe ◽  
Ali H. M. Murid
Keyword(s):  
Mathematical Model ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Fourth Order ◽  
Difference Method ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Disinfectant Solution ◽  
Surface Decontamination ◽  
Runge Kutta

A mathematical model is considered to determine the effectiveness of disinfectant solution for surface decontamination. The decontamination process involved the diffusion of bacteria into disinfectant solution and the reaction of the disinfectant killing effect. The mathematical model is a reaction-diffusion type. Finite difference method and method of lines with fourth-order Runge-Kutta method are utilized to solve the model numerically. To obtain stable solutions, von Neumann stability analysis is employed to evaluate the stability of finite difference method. For stiff problem, Dormand-Prince method is applied as the estimated error of fourth-order Runge-Kutta method. MATLAB programming is selected for the computation of numerical solutions. From the results obtained, fourth-order Runge-Kutta method has a larger stability region and better accuracy of solutions compared to finite difference method when solving the disinfectant solution model. Moreover, a numerical simulation is carried out to investigate the effect of different thickness of disinfectant solution on bacteria reduction. Results show that thick disinfectant solution is able to reduce the dimensionless bacteria concentration more effectively

Application of the piecewise constant method in nonlinear dynamics of drill string

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Jialin Tian ◽  
Jie Wang ◽  
Yi Zhou ◽  
Lin Yang ◽  
Changyue Fan ◽  
...  
Keyword(s):  
Nonlinear Dynamics ◽  
Dynamic Model ◽  
Dynamic Characteristics ◽  
Drill String ◽  
Vibration Velocity ◽  
Kutta Method ◽  
Time Step ◽  
Piecewise Constant ◽  
Runge Kutta Method ◽  
Runge Kutta

Abstract Aiming at the current development of drilling technology and the deepening of oil and gas exploration, we focus on better studying the nonlinear dynamic characteristics of the drill string under complex working conditions and knowing the real movement of the drill string during drilling. This paper firstly combines the actual situation of the well to establish the dynamic model of the horizontal drill string, and analyzes the dynamic characteristics, giving the expression of the force of each part of the model. Secondly, it introduces the piecewise constant method (simply known as PT method), and gives the solution equation. Then according to the basic parameters, the axial vibration displacement and vibration velocity at the test points are solved by the PT method and the Runge–Kutta method, respectively, and the phase diagram, the Poincare map, and the spectrogram are obtained. The results obtained by the two methods are compared and analyzed. Finally, the relevant experimental tests are carried out. It shows that the results of the dynamic model of the horizontal drill string are basically consistent with the results obtained by the actual test, which verifies the validity of the dynamic model and the correctness of the calculated results. When solving the drill string nonlinear dynamics, the results of the PT method is closer to the theoretical solution than that of the Runge–Kutta method with the same order and time step. And the PT method is better than the Runge–Kutta method with the same order in smoothness and continuity in solving the drill string nonlinear dynamics.

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