# A Numerical Analysis of the Influence of Oxygen and Glucose in Healthy and Tumour Cells
###### Abstract
In the case of healthy cells, similar to what occurs under hyperoxia, excessively high glucose concentrations are toxic and stimulate the production of reactive oxygen species, resulting in physiological and pathophysiological changes as well as cellular and DNA damage. Prolonged exposure to this condition (1-13 days) can disrupt normal mitochondrial functions and contribute to the development of pathological conditions such as cardiovascular diseases, cancer inflammation, and immune system disorders [26; 27; 28; 29]. This may inhibit cell proliferation, induce growth arrest, or trigger cellular apoptosis [30]. Due to the complex behaviour of cells during the process of proliferation and the vast number of interconnected processes and molecules involved during the process, it is considered a complex, multiscale phenomenon that can be challenging to modulate [31; 32]. Currently, mathematical and computer models are frequently employed to study various aspects of cancer biology, carcinogenesis, and tumour proliferation [1]. These models have emerged as platforms for exploring different hypotheses, developing new therapeutic strategies, and optimising treatment effectiveness, prevention, and diagnosis [33; 34]. However, constructing a comprehensive 3D model that incorporates tissue architecture, proliferation, angiogenesis, invasion, and the fundamental cellular/microenvironmental features remains a significant challenge in the field [33]. To solve these models, numerical methods are commonly employed, and various types of methods have been used in the field of cell proliferation [35; 36; 37; 38]. The finite element method is currently the most widely used method across different areas. However, it was not until the 1980s that it started being used for studying biomedical problems [39]. This method is known for its simplicity but relies on a mesh that discretizes the domain. In complex problems involving large deformations, such as cell proliferation, this characteristic can be a drawback since the mesh can impact the results obtained. Mesh distortion reduces the precision and robustness of the final result. In order to tackle it, the meshless method emerged [40]. These methods involve discretizing the domain with a nodal distribution without establishing relationships between the nodes, in contrast to a mesh where nodes are interconnected [41; 42]. Within meshless methods, they can be classified as approximation and interpolation meshless methods. The approximation methods utilise approximate functions, aiming to deliver smoother solutions and obtain the influence domains using a fixed radial search. In these approaches, the integration mesh is independent of the nodal arrangement [43]. However, one of their main disadvantages is that the shape functions they create do not possess the delta Kronecker property, making it challenging to directly impose boundary conditions, which increases computational costs. Examples of these methods include the diffuse element method, the element-free Galerkin method, smoothed particle hydrodynamics, and the meshless local Petrov-Galerkin method [41]. On the other hand, the interpolation meshless methods are known for possessing the delta Kronecker property, eliminating the disadvantage mentioned above. For this reason, the use of these methods is generally preferred [42]. In the formulation of meshless methods, the domain is discretized using an unstructured set of nodes, then the nodal connection is established and the shape functions are created. Since elements are not utilised, the resulting shape functions possess a virtually higher order, leading to more accurate approximations and allowing for a more refined process through the easy addition or removal of nodes [41]. As a result, meshless methods have emerged as powerful numerical tools for efficiently solving complex problems [40]. An example of an interpolation method is the radial point interpolation method (RPIM) [44; 45].