## 2 Mathematical Modeling  
In cancer research, mathematical modeling plays a pivotal role in comprehending the intricate dynamics of cancer in response to various treatments [23; 24]. It complements cancer research by providing valuable quantitative predictions to unravel the complexity of cancer and open new avenues for developing effective treatments [25]. Numerous mathematical models, often rooted in ordinary differential equations (ODEs), have explored various aspects of cancer research, including treatment strategies [26; 27; 28; 29; 30; 31; 32; 33; 34], immune responses [35; 36; 37; 38; 39; 40; 41], treatment sensitivity and resistance [42], the effects of treatment combinations [43; 44; 45; 46; 47; 48; 49], habitat dynamics [50], tumor heterogeneity [51; 52], and treatment optimization [53]. Their collective contribution lies in furnishing predictions for optimal dosing, treatment regimens, and scheduling, all while minimizing adverse side effects and maximizing therapeutic gains. This avenue holds promise for personalized cancer care, customizing treatments based on individual patient traits and diverse tumor characteristics, including size, type, growth rate, heterogeneity, genetics, and immune composition [37; 54]. For mathematical models to operate with precision, the accurate identification of critical parameters hinges on the acquisition of frequent and precise data [55]. ODE-based equations are commonly employed in mechanistic and deterministic models where underlying mechanisms and deterministic relationships are of primary interest (Figure 1). Beyond the realm of ODEs, fractional calculus emerges as a potential frontier for refining cancer modeling, particularly in capturing memory effects and intricate aspects of tumor growth [56; 57; 58]. Fractional calculus enables the differentiation and integration of non-integer orders, extending beyond traditional calculus. Cancer research finds application by modeling anomalous diffusion processes, characterizing tumor behavior more accurately, and incorporating memory effects into models, deepening our understanding of tumor dynamics [59; 60; 61; 62]. Several mathematical models have been validated through preclinical investigations and clinical trials, demonstrating their potential for shaping effective clinical trial designs [63; 64; 65; 66; 67; 68; 69]. Notably, [PERSON] et al. [69] engineered a mathematical model tailored to refine radiation therapy schedules for glioblastoma, a formidable brain cancer. The model recommended an almost-optimal re-irradiation regimen for recurrent glioblastoma patients: 7 days of 3.96 Gy (administered once per day), followed by 9 days of 1.0 Gy (administered thrice per day). Strikingly, this regimen demonstrated both feasibility and safety in all 14 patients (100%) who received it, marking a notable achievement in glioblastoma treat ment. Although preliminary results hint at enhanced tumor control, the imperative for larger-scale trials to substantiate its efficacy remains paramount. These findings prove the invaluable role of mathematical modeling in fine-tuning cancer treatment strategies, offering promising prospects for future clinical applications. After the proposal of adaptive and self-adaptive plasma [70], an exponential growth model was developed [71] to predict the dynamic response of cancer cells (U-87 MG and MDA-MB-231) under CAP treatment [18]. This model successfully derived optimal treatment conditions for two distinct cancer cell lines: U-87 MG and MDA-MB-231. For U-87 MG, the study aimed for a desired ratio of cancer cell viability (rd) of 0.5, leading to optimal treatment parameters of \(\Delta\)t (treatment duration) at 65.1 s and U (Plasma Dis  
Figure 1: Mathematical models/approaches employed for the application of CAP in oncology, highlighting their respective advantages and disadvantages. charge Voltage) at 3.367 kV. For MDA-MB-231, targeting an rd of 0.3 resulted in optimal settings of \(\Delta\)t at 71.6 s and U at 3.315 kV.