## 1 Introduction  
In many societies, cancer is the leading cause of death and a serious health burden. Different anticancer and chemotherapy medicines have been created to combat these risky diseases [1]. The majority of chemotherapy drugs do not have a selective effect on cancer cells and can destroy healthy cells as well. As a result, supplying the medication directly to the tumor without affecting healthy tissues is a significant obstacle in cancer care [2, 3]. Various attempts, such as \"smart drug delivery\" using nano/submicron droplets [4], have been made to resolve this problem. Researchers looking into smart drug distribution are having trouble figuring out how to get the drug to the right place at the right time. Drug delivery is one of the most focused research areas for treating such complex diseases as cancerous tumors in various body organs such as the brain, breast, lungs, kidney, and heart. Nanoparticles such as iron oxide are used in nanomedicine to diagnose and treat diseases, and MRI imaging with iron oxide was recently used to track drug delivery in mice. Bacteria were used to transport drugs to tumor sites [5], with chemoattractant gradients controlling the transport path. Chemotaxis is a basic cell and organism guidance system that attracts microbes to substance, immune cells to infection sites. The Patlak-Keller-Segel (PKS) method is a good choice for modelers and analysts alike, and it is a portion of the capstone of biological modeling. Chemotaxis is the tendency of cells and living organisms to move in response to an external stimulus in their environment. Due to its critical function in cancer metastasis, embryogenesis, cell growth, inflammation, and wound healing [6], chemotaxis has been extensively studied in the last two decades [7; 8]. Macrophages are known to migrate to areas of inflammation, which are typically where hypoxic tumors are found. In [9], a mathematical model of macrophage migration toward hypoxic tumor sites is suggested as a drug delivery vehicle. To demonstrate macrophage penetration into tumors, the authors propose a rough modeling process. They also look into the action and development of chemotaxis and chemokines, as well as the efficacy of drug-loaded macrophages for drug delivery. A growing avascular tumor spheroid serves as the basis for the model. Chemotaxis is a subset of the reaction-diffusion systems method. Through self-organizing/pattern formation, reaction-diffusion systems create a wide range of spatiotemporal patterns, including propagating circular, spiral, and periodic waves. For the first time, such spatiotemporal patterns are observed in the Belousov-Zhabotinsky reaction system [10]. A standard model for explaining the pattern-forming mechanism is the reaction-diffusion system. Indeed, [PERSON] developed in his seminal paper [11] the idea that chemicals can react and diffuse in such a way that they create steady-state nonhomogeneous spatial patterns of chemical concentrations under certain conditions. Although a diffusion process generally brings uniform distribution of a substance, the scenario of [PERSON] presents a non-uniform pattern as a stationary state in a reaction-diffusion system. We can find a wide variety of reaction-diffusion systems in the fields of physics, chemistry, and biology [12]. One of the most popular reaction-diffusion systems is the chemotaxis model. Chemotaxis is the feature movement of a cell along a chemical concentration gradient either toward the chemical stimulus, in which case the chemical is called chemoattractant, or away from the chemical stimulus, in which case the chemical is called chemo-repellent. Theoretical and mathematical modeling of chemotaxis dates to the pioneering work of [PERSON] in the 1950s [13] and [PERSON] in the 1970s [14; 15]. The review article by [PERSON] [16] provides a detailed introduction into the mathematics of the Keller-Segel (KS) model for chemotaxis. The general form of the Keller-Segel model in which we are interested is given by the system  
\[\left\{