Mohammed Abdulhameed

In the search for causes and cures of cancer diseases, many mathematical models developed have resulted in systems of nonlinear stiff ordinary differential equations. With these models, many numerical estimates of biological knowledge of the parameters have been obtained, a number of phenomena interpreted, and predictions were made in order to gain further knowledge of cancer development and possible treatment. In this study, numerical simulations of the models were performed using continuous block implicit hybrid methods and the results obtained support the theoretical and clinical findings. We analyzed the interactions among the various tumor cell populations and present the results graphically. From the graphical representation of results, one can clearly see the effects of all the tumor cell populations involved in the competition, as well as the effects of some treatments by the applications of some therapeutic agents which have been heavily used in the clinical treatments of breast cancer. The treatments in the past were mostly conventional chemotherapies, which were used either singly (alone) or in combination with each other or other therapies, and all played vital roles, except for the side effects that these therapies incur in normal tissues and organs. Thus, from recent research works, it is now clear that in many cases they do not represent a complete cure. Therefore, the need to address not only the preventative measures of breast cancer, but also more successful treatment, is clear, and can be successfully achieved to increase the survival rate of breast cancer patients.

The combined influence of magnetic field, electro-osmotic flow (EOF), and pressure gradient on the unsteady magnetohydrodynamic fluid flow through a parallel microchannel is investigated. The Burger's liquid model is used for the fractional partial differential equation, which allows us to study the behavior of viscoelastic liquid velocity profile in the parallel microchannel. The Laplace transform (LT) in concert with the Fourier cosine transform are used to obtain the analytical solution of the velocity profile. Furthermore, using the method of separation of variables, the energy equation, Joule heating, energy dissipation, and electromagnetic effects are calculated to obtain the temperature within the microchannel. The influence of some relevant parameters like heat transfer, temperature distribution, Hartmann number (Ha), and Brinkman number (Br) on fluid flow velocity are presented graphically and discussed. The results obtained show that temperature decreases with Ha and Br of the transverse electric field for slip flow. Close to the middle of the microchannel, fluid flow velocity increases with decrease in the delay time parameter value and increase in the Burger's parameter value, while the opposite trend is found for the velocity near to the middle of the microchannel. In addition, Burger's liquid is quite general, such that Oldroyd-B, Maxwell, and Newtonian liquids are readily obtained as limiting cases. The study is significant in the application fields of chemical analysis and biological analysis, drug delivery, bacteria detection, and some others.

The differential equations of pharmacokinetic models, obtained from the formulation based on the Fick's perfusion principle and law of conservation of mass action, deals with absorption, distribution, and elimination of drugs by the body systems. In the formulation, the ordinary differential equations obtained may result into highly stiff systems whereby suitable implicit methods have to be employed to solve accurately drug concentration in the body systems, owing to the complex nature of the exact solutions, if they exist. Further, because of the large and increasing interest in the problems of drug kinetics, it is necessary to apply considerably accurate implicit numerical methods in evaluating and in applying them to specific cases. These solutions are to help determine the distribution of drug concentration in different compartments (parts) of the human body network systems as a fundamental step to help in understanding and improving treatments. This is necessary since drug concentration in each compartment is different from one another because of the differences in drug affinity to tissues. From the study, the solution curves obtained show that drug level in the gastrointestinal tract decreases with the passage of time, while drug concentration in the blood increases from zero and reaches its maximum level and then decreases steadily again. We further, compared the model curves with some experimental data plots published in scientific papers. The results obtained from the study can be used extensively for various drug diffusion problems arising in pharmaceutical studies. The presented results widen the applicability of the continuous block implicit hybrid one-step collocation methods to diffusion process which have good number of applications in the drug control, drug dosage and other related problems in pharmaceutical and biomedical industries.

In this paper, we construct the exact solution of blood flow of Oldroyd-B fluids to represent the non-Newtonian characteristics of biofluids and to study the unsteady flow of blood and heat transfer through arterial segment with external magnetic field applied normal to the flow direction in the presence of thermal radiation and body acceleration. In the study, we investigate the influence of Caputo time-fractional parameter, heat transfer, external body acceleration, and magnetic field on the blood velocity and temperature distribution in a straight circular cylindrical vessel. The constitutive partial differential equation and temperature equation governing blood flow in the arterial wall are solved using Laplace and finite Hankel transforms. In addition, Gaver Stehfest’s algorithm is used for the inverse Laplace transform. The results obtained are interpreted graphically and discussed from the physiological point of view. The graphs show that the rate of heat transfer and blood velocity become less with the increase in the external magnetic field strength which is very good in regulating blood flow as well as temperature distribution during treatment. Based on the graphical representations, blood velocity and temperature distribution both decrease in ascending order of the fractional parameter as memory effect. Further, owing to the dissipation of energy caused by blood viscoelasticity and magnetic field effect, during pulsatile flow of blood, the heat transfer rate at the wall of the artery is enhanced. The significance of this study can be found in the application fields such as biomedical engineering, medicine and pathology.

In this paper, we construct transient electro-osmotic flow of Burgers’ fluid with Caputo fractional derivative in a micro-channel, where the Poisson–Boltzmann equation described the potential electric field applied along the length of the microchannel. The analytical solution for the component of the velocity profile was obtained, first by applying the Laplace transform combined with the classical method of partial differential equations and, second by applying Laplace transform combined with the finite Fourier sine transform. The exact solution for the component of the temperature was obtained by applying Laplace transform and finite Fourier sine transform. Further, due to the complexity of the derived models of the governing equations for both velocity and temperature, the inverse Laplace transform was obtained with the aid of numerical inversion formula based on Stehfest's algorithms with the help of MATHCAD software. The graphical representations showing the effects of the time, retardation time, electro-kinetic width, and fractional parameters on the velocity of the fluid flow and the effects of time and fractional parameters on the temperature distribution in the micro-channel were presented and analyzed. The results show that the applied electric field, electro-osmotic force, electro-kinetic width, and relaxation time play a vital role on the velocity distribution in the micro-channel. The fractional parameters can be used to regulate both the velocity and temperature in the micro-channel. The study could be used in the design of various biomedical lab-on-chip devices, which could be useful for biomedical diagnosis and analysis.

We consider the unsteady flow of Burger fluid within a circular cylindrical tube, driven by a time-dependent pressure gradient, a body acceleration and a magnetic field acting normal to the flow direction. The solutions of the fractional constitutive equations governing the unsteady Burger’s fluid flow through arterial walls were obtained via the Laplace transform and the finite Hankel transform. The effects of magnetic field on parameters such as blood temperature and velocity were studied by Caputo time-fractional derivatives. We note that the solutions of many particular models such as fractional Oldroyd-B fluid, fractional Maxwell fluid, fractional second grade fluid and fractional Newtonian fluid models can be recovered from the solutions of the fractional constitutive equations governing the unsteady Burger’s fluid flow by particularizing the material coefficients (i.e. special and limiting cases of the earlier Burger’s fluid model). The numerical computations have been carried out to analyze the effects of fractional parameter α, similarity parameter β, relaxation time λ₁, retardation time λ₃, radius of the circular cylinder R₀ and material parameter λ₂ on the blood velocity and temperature. Some interesting flow and temperature characteristics are presented graphically and discussed. The study reveals that blood velocity, temperature and fractional parameters are reduced in the presence of magnetic field. The importance of this study can be found in the application fields of magnetic field control of biotechnological processes, bio magnetic device technology, biomedical engineering, medicine, etc.