Mathematical Modeling for Brain Tumors Including Fractional Operator 165

is motivated by their ease. Their relative simplicity makes it possible to de-

rive analytical solutions, which may be used to analytically characterize the

evolution of events.

Additionally, the flexibility of ODE-based models and the ability to fine-

tune their free parameters versus experimental data to represent various tu-

mor phases make them a good choice for supporting therapeutic recommenda-

tions. Nevertheless, there is disagreement about whether ODE-based model

is best suited for particular types of cancer. An inappropriate model selec-

tion may end up in significant variations in cancer predictions, indicating the

need for more study. Thus, ODE models for tumor development may be ex-

panded so that complexity is contained in simplicity, while maintaining their

deductive-reductionist features to better suit experimental data. Fractional

calculus examines integral calculus and non-integer order differential among

other mathematical options. The ability to handle unique behavior according

to an arbitrary degree of differentiation (or integration) is a characteristic

shared by fractional models, which broadens the application scope. Even in

comparatively smaller models, this is a crucial feature of non-integer order cal-

culus that makes it a fascinating instrument for reductionist methods. Addi-

tionally, the fundamental attributes of fractional calculus, such as its capacity

to indicate complex procedures like long-term memory and/or geographic het-

erogeneity, may enhance ODE-based tumor models. Fractional models have

gained popularity and been effectively used in a variety of fields, including

signal processing, thermoacoustics, economics, robotics, viscoelasticity, chem-

ical kinetics, electromagnetism, agricultural computing, and traffic control,

attributed to their many outstanding advantages [35–38]. As a matter of fact,

there is already what could be called “fractional mathematical oncology”—a

number of recent studies have used non-integer calculus to deal with vari-

ous aspects of cancer, such as the dynamics of chemotherapy, radiotherapy,

and immunotherapy, as well as numerical solutions and control for invasion

structures, bioengineering, and tumor growth [39–42]. As more research is

conducted in this field, fractional calculus may be supported as an alternative

reductionist phenomenological modeling (method) to study early avascular

general tumor growth controlled by ODEs [43–50]. The general form of tumor

development models driven by ODE equations are usually the following:

dV (t)

dt

= af(V (t)) −bg(V (t)),

(6.3)

where V (t) is tumor volume at a given time and t, ^{dV (t)}

dt

represents tumor

growth rate. Tumor growth is determined by a parameter called a, whereas

tumor size is constrained by a parameter called b. Functions f(V (t)) and

g(V (t)) define whether or not the model is logistic, exponential, or in some

other form.

For certain ordinary differential equations, models, differential equations,

maximum size, and growth conditions are provided, respectively in [51]: