The growth and development of the brain continue for several years after birth, largely due to the presence and activity of glial cells. These cells provide essential physical and chemical support to neurons, help maintain the neural environment, and play a critical role in the central nervous system (CNS). However, when glial cells grow uncontrollably, they can disrupt normal brain development and function, leading to various CNS disorders. One such disorder is glioma, a type of tumor caused by the over-proliferation of glial cells. When glioma occurs in children, it is referred to as pediatric glioma, typically treated through a combination of chemotherapy, radiotherapy, and surgical intervention.
In recent years, mathematical modeling has emerged as a powerful tool for understanding the complex biological dynamics of the brain and its disorders. These models allow researchers to analyze biological problems, formulate and test hypotheses, and gain deeper insights into the mechanisms underlying disease progression and treatment response. A foundational mathematical formulation of the glioma model was introduced by Wein and Koplow, governed by a second-order partial differential equation.
This study focuses on a modified glioma model that incorporates a treatment parameter to account for therapeutic effects. The model describes the spatiotemporal evolution of glioma density in the absence of treatment and includes a nonlinear term to model the suppression of glioma growth under treatment.
To evaluate the effectiveness of various semi-analytical methods in solving nonlinear partial and fractional differential equations relevant to glioma dynamics, three methods were employed: the Homotopy Analysis Method (HAM), Homotopy Perturbation Method (HPM), and Reduced Differential Transform Method (RDTM). These methods were used to derive approximate solutions to the modified glioma model, aiming to capture the approximate linear progression of glial cell concentration during treatment.
The approximate solutions generated by these methods were illustrated in figures, providing a basis for comparative analysis of their convergence behavior and accuracy. The figures also showcased the linear growth profile produced by RDTM at various time points, emphasizing its rapid stabilization relative to HAM and HPM. The influence of medical treatment parameters on the spatial spread of glial cells was illustrated in another figure, where the radius of cell concentration was shown to decrease over time.
The study further examined the role of fractional derivatives in shaping the concentration dynamics of glioma cells by extending the model to a time-fractional reaction-diffusion framework. This extension enabled the incorporation of memory effects and anomalous diffusion behaviors frequently observed in biological systems. To solve the fractional model, the Homotopy Perturbation Method (HPM) was implemented using two distinct strategies: direct recursive expansion and an embedding parameter formulation involving Mittag-Leffler functions. Additionally, the Fractional Reduced Differential Transform Method (FRDTM) and the Reduced Differential Transform Method (RDTM) were employed to provide a consistent and comparative evaluation of solution performance within this extended context.
The analysis was grounded in the Caputo-type fractional diffusion equation, which served as the mathematical foundation for modeling the time-dependent behavior of glioma cell concentration under fractional-order dynamics. To ensure the reliability of the obtained solutions, a thorough investigation of their convergence properties was conducted, and error estimates were provided for all series solutions. The impact of varying fractional orders was illustrated in figures, which displayed solution profiles and line graphs capturing the system's evolving dynamics.
In addition to the numerical analysis, the existence, uniqueness, and continuous dependence of the solution to the fractional model were established through three complementary analytical approaches: the fixed-point method via the fractional Volterra formulation, the spectral semigroup approach involving Mittag-Leffler functions, and the Laplace transform technique. Each method verified that the problem was well-posed within suitable functional spaces. These theoretical properties were visually supported in a figure, which illustrated the stability and structure of the solution space under the fractional framework, reinforcing the mathematical soundness of the model.
The study concluded that the three analytical methods HAM, HPM, and RDTM were effective in deriving approximate solutions to the nonlinear partial differential equation representing the treated glioma model. RDTM stood out for its superior efficiency and accuracy, particularly in capturing the early-time dynamics of the system. The analysis of the glioma cell radius showed a consistent decline over time, emphasizing the impact of treatment parameters in suppressing glioma expansion. By extending the model to its fractional form, the existence and uniqueness of the solution were established, and the behavior of the three-dimensional solution profiles revealed a steady decrease in glial cell concentration, ultimately reaching zero, indicating complete growth suppression. Among the fractional parameters tested, α = 0.5 proved most effective in accelerating the decline, highlighting the significance of fractional derivatives in enhancing model precision and therapeutic insight.
