Mathematical Modeling for Brain Tumors Including Fractional Operator 167

6.6

Mathematical Modeling of Brain Tumors Using

Fractional Operator

Fractional differential equations, fractional integro-differential equations, frac-

tional partial differential equations, and so on are variant classes of fractional

equations that have been applied recently in the areas of control, mechan-

ics, chemistry, physics, biology, signal processing, and economics to model

various real phenomena like viscoelasticity and damping, diffusion and wave

propagation, control systems, and signal processing [34]. As a consequence, a

great deal of research has been done by several scientists in the domains of

biology, engineering, physics, and other sciences to illustrate many significant

phenomena in these subjects via the FC. These can be seen in [52–59]. For

instance, the authors have provided a fractional operator model for the co-

infection of HIV and HSV-2. Furthermore, a Covid-19 mathematical model is

worked out. A COVID-19 fractional model was utilized in Galicia, Spain, and

Portugal. Study was done on a fractional model of the tumor-immune system

using fractional derivative and a mathematical model of Parkinson’s disease

in the basal ganglia. A fractional operator is used to describe the growth of

brain tumors (glioblastomas) during medical therapy [51]. The model was ini-

tially built to resemble a case of chemotherapy-treated recurrent anaplastic

astrocytoma. Later, it was adjusted to enable assessment of the impacts of

surgical resection extent as well as growth and diffusion changes to cover the

complete spectrum of glioma behavior. Although this glioma can develop at

any age, individuals over 45 are more likely to get glioma. They can occur

in the cerebellum as well as the cerebral hemispheres, where they typically

occur. A number of researchers have examined the mathematical equation of

tumor growth and have provided explanations of the basic two-dimensional

tumor model.

In mathematical words, rate of change of tumor cell density = diffusion of

tumor cells + growth of tumor cells,

∂τB(x, τ) = D∇²B(x, τ) + ρB(x, τ)

(6.4)

The two major processes in the development of an invading brain tumor

are taken into account in these models, cell proliferation (ρ) and diffusion (D),

and combines them to create an equation for reaction and diffusion:

∂τB(x, τ) = D ¹

x^{2 ∂x(x2∂xB(x, τ)) + ρB(x, τ)}

(6.5)

The tumor cells in these models are considered to grow at a constant pace,

exponentially. Owing to the significance of these models, it can sometimes be

difficult to obtain the results. As a consequence, there is a great deal of focus

on developing methods for estimating solutions associated with these models.

A number of researchers analyzed a patient’s final year of CT images to create