A Numerical Analysis of the Influence of Oxygen and Glucose in Healthy and Tumour Cells

Abstract

Cancer is widely regarded as a critical health issue in modern society. Tumour cells are usually associated with abnormal proliferation that disrupts the normal behaviour of the body. All cells depend on the availability of oxygen and nutrients present in the extracellular environment, which can enhance or decrease their ability to proliferate. Therefore, to comprehend the influence of these factors, it is helpful to understand the proliferation process of both healthy and tumour cells. Computational models are powerful tools used to study biomedical problems, and several models have been presented in the literature. Different numerical methods have been proposed to solve these models. Among them, meshless methods can be highlighted, as they are used to solve complex problems with accurate results. However, in the case of cell proliferation, this is still an area that has not yet been fully explored. The aim of this work is to implement and study the influence of oxygen and glucose during the proliferation of healthy and tumour cells using a novel algorithm. This is an iterative discrete algorithm that employs a meshless numerical and uses a new phenomenological law to describe cell growth. In the end, the algorithm was capable of generating satisfactory results, in accordance with the literature.

1. Introduction

Cancer is one of the deadliest diseases of this century [1,2,3] and remains a significant public health problem worldwide [4]. In 2018 alone, a staggering 18.1 million people were affected by this condition, resulting in 9.6 million deaths [3].

This disease is characterised by the emergence of genetic mutations or epigenetic changes that disrupt the normal behaviour of individual cells or cell populations. These changes can be inherited by subsequent cell generations and even give rise to new mutations. Such mutations enable cells to evade internal controls and often promote abnormal growth, division, and proliferation [1].

Abnormal proliferation is a major hallmark of cancer, leading to imbalances between cell proliferation and cell death. This compromises various tissues, causing morbidity and ultimately leading to death [5].

When tumour cells begin to proliferate, they depend on oxygen, nutrients, and growth factors obtained from the vasculature of the host tissue through diffusion [1]. Typically, the extracellular environment provides the necessary nutrients for cell proliferation and does not significantly influence the process under normal conditions [6]. However, when these conditions are disrupted, this process can be compromised [7].

Regarding oxygen, it is essential for both healthy and tumour cells. Oxygen has a significant impact on cell proliferation, as cells require adequate oxygenation for proper metabolic processes and cellular functions to occur [8,9,10]. When oxygen concentrations exceed normal levels (i.e., hyperoxia), cells experience a toxic effect. Under such conditions, cells are prone to oxidative damage caused by the production of reactive oxygen species, which can induce apoptosis and necrosis depending on the duration and level of hyperoxic exposure [11,12]. Under hypoxia (i.e., low oxygen concentrations), the rate of cell proliferation decreases, and the process of apoptosis can be activated to prevent the emergence and progression of mutations in subsequent cell generations, thereby attempting to prevent the formation of tumour cells [13,14]. When this condition becomes severe, respiration is impaired, and cells die within a short period, typically ranging from 10 to 48 h [15]. In the case of tumour cells, hypoxia influences tumour progression, aggressiveness, metastasis, and resistance to apoptosis, treatments, and radiation [9,16,17]. However, hypoxia alone is usually insufficient to neutralise tumour cells [17,18]. Initially, cells tend to decrease their proliferation rate; however, this leads to excessive resource consumption and abnormal angiogenesis, which, in turn, promote tumour motility in search of a new oxygen source, ultimately increasing the proliferation rate [18]. However, similar to what occurs in healthy cells, when this condition becomes severe and extreme, tumour cells stop proliferation and die [7,19]. Within 24 h, cell proliferation may stop, and approximately 20% to 30% of the affected cells are prone to die under these conditions [17].

On the other hand, hypoxia can also increase glucose consumption in tumour cells [16]. When comparing the metabolism of tumour cells to healthy cells, tumour cells tend to consume glucose at higher rates [20,21]. Glucose serves as the primary molecule utilised in their metabolism, facilitating their proliferation [22,23]. Therefore, under conditions of glucose deprivation, tumour cells tend to halt their proliferation due to ATP depletion [7], leading to metabolic disturbances, oxidative stress, alteration of the redox status, and cellular death [24,25]. 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].

The focus of this study is the implementation and study of the impact of oxygen and glucose on the proliferation of both healthy and tumorous cells, using an algorithm developed by the authors [46]. This algorithm, which has been previously created and simulates cell proliferation, is integrated with the RPIM. Although this method has never been employed to analyse this specific biological issue, the authors believe it has potential due to its demonstrated capabilities in [46]. Furthermore, it offers advantages in comparison to other meshless methods as it permits the utilisation of structures already defined in various accessible FEM softwares.

Furthermore, the phenomenological law introduced in [46], which outlines cell growth up to the division phase, has been integrated into the algorithm. The growth law was adapted to rely on the concentrations of oxygen and glucose. Throughout the iterations, these concentrations are computed independently by solving the respective reaction–diffusion system of equations. Moreover, for each molecule, five concentration ranges are defined: extreme hyperoxia, hyperoxia, normal oxygen concentration, hypoxia, and extreme hypoxia for oxygen, and extreme hyperglycaemia, hyperglycaemia, normal glucose concentration, hypoglycaemia, and extreme hypoglycaemia for glucose. Each concentration range has a specific influence on cell growth. The different states defined for these two molecules result in twenty-five different possibilities. For each combination, a growth curve, representing the cell volume over time, was obtained. Therefore, this study implements and examines the evolution of the cell when exposed to normal, abnormal, and extreme concentrations of glucose and oxygen.

2. Materials and Methods

After defining the geometry of the domain and determining its limits, as well as identifying the natural and essential boundary conditions (Figure 1a), a generic meshless procedure can be followed. Initially, the domain of the problem is discretized with a nodal distribution, denoted as (nodes): $\mathit{N}=\left\{n_0,n_1,\dots , n_N\right\}$, which can be either regular or irregular. (Figure 1b). Since these nodes are independently scattered without any established relationship between them, the discretization technique is referred to as meshless [47]. Subsequently, a background integration grid is established, and integration points are constructed, representing material points. The assembly of all integration points represents the complete volume of the domain. These integration points are necessary to perform the numerical integration of the integro-differential equations governing the physical phenomena under study. Next, nodal connectivity is defined. Each integration point searches for its closest neighbouring nodes, establishing influence domains for all integration points. When these influence domains are overlapped, nodal connectivity is established. All these mathematical structures are required to construct the shape functions, which are developed using the radial point interpolating technique. Using the shape functions, the integro-differential equations are formulated, and the resulting system of equations is established and solved. A detailed description of the main steps is provided in the following sections.

2.1. Numerical Integration

After the domain is discretized, the next step is to define integration points that will represent the volume of the analysed domain. This integration mesh can be constructed using various techniques. To obtain this mesh, the domain can be divided into triangular or quadrilateral integration cells (in this work, quadrilateral integration cells were used) (Figure 1c). The Gauss–Legendre quadrature scheme was assumed to determine the placement of integration points within each of these cells (Figure 1d). Subsequently, the integration cells are transformed into isoperimetric squares, and the Gauss–Legendre quadrature rule is used to distribute the $n_Q\times n_Q$ integration points within each square. Ultimately, the isoparametric interpolation functions are used at these points to obtain their Cartesian coordinates. A comprehensive description of this step can be found in the literature [47,48].

2.2. Nodal Connectivity

Once the integration mesh is obtained, the next step is to establish nodal connectivity. This is achieved by performing a radial search for each integration point ${\mathit{x}}_I$ in order to identify a predetermined number of nearby nodes n near the considered point (Figure 1b). Note that this search is centred on ${\mathit{x}}_I$. The nodes that are found for each ${\mathit{x}}_I$ form its “influence domain”, meaning that all integration points have a specific influence domain. Then, these influence domains are naturally overlapped, and so the nodal connectivity is established. The size and shape of the influence domains affect the accuracy of the results [47,49,50]. Thus, for all influence domains, as suggested in the literature, seven nodes were considered.

2.3. Shape Function

To obtain the interpolation shape functions, the RPIM resorts to the RPI technique, which is a combination of a polynomial basis function with a radial basis function [44,47]. A field variable $u\left(\mathit{x}\right)$ can be interpolated at an integration point ${\mathit{x}}_I$ with:

$$u\left({\mathit{x}}_I\right)=\sum _{i=1}^n{R}_i\left({\mathit{x}}_I\right)·a_i\left({\mathit{x}}_I\right)+\sum _{j=1}^m{p}_j\left({\mathit{x}}_I\right)·b_j\left({\mathit{x}}_I\right)={\mathit{R}}^T\left({\mathit{x}}_I\right)\mathit{a}\left({\mathit{x}}_I\right)+{\mathit{p}}^T\left({\mathit{x}}_I\right)\mathit{b}\left({\mathit{x}}_I\right)$$

(1)

Or as:

$$u\left({\mathit{x}}_I\right)=\left\{{\mathit{R}\left({\mathit{x}}_I\right)}^{\mathrm{T}},{\mathit{p}\left({\mathit{x}}_I\right)}^T\right\}\left\{\begin{array}{c}\mathit{a}\left({\mathit{x}}_I\right)\\ \mathit{b}\left({\mathit{x}}_I\right)\end{array}\right\}$$

(2)

where $n$ and $m$ are the number of nodes of the influence domain on the point ${ \mathit{x}}_I$, $\mathit{R}\left({\mathit{x}}_I\right)$ the radial basis function, $\mathit{p}\left({\mathit{x}}_I\right)$ the polynomial basis function, and $a\left({\mathit{x}}_I\right)$ and $\mathit{b}\left({\mathit{x}}_I\right)$ the non-constant coefficients of each function, respectively. Moreover, considering $\mathit{R}\left({\mathit{x}}_I\right)$,$\mathit{p}\left({\mathit{x}}_I\right)$, $\mathit{a}\left({\mathit{x}}_I\right)$ and $\mathit{b}\left({\mathit{x}}_I\right)$, these matrices are obtained through the following expressions:

$$\mathit{R}\left({\mathit{x}}_I\right)={\left[R_1\left({\mathit{x}}_I\right)R_2\left({\mathit{x}}_I\right)\dots R_n\left({\mathit{x}}_I\right)\right]}^{\mathrm{T}}$$

(3)

$$\mathit{p}\left({\mathit{x}}_I\right)={\left[p_1\left({\mathit{x}}_I\right)p_2\left({\mathit{x}}_I\right)\dots p_m\left({\mathit{x}}_I\right)\right]}^{\mathrm{T}}$$

(4)

$${\mathit{a}}^{\mathrm{T}}\left({\mathit{x}}_I\right)=\left[a_1\left({\mathit{x}}_I\right)a_2\left({\mathit{x}}_I\right)\dots a_n\left({\mathit{x}}_I\right)\right]$$

(5)

$${\mathit{b}}^{\mathrm{T}}\left({\mathit{x}}_I\right)=\left[b_1\left({\mathit{x}}_I\right) b_2\left({\mathit{x}}_I\right)\dots b_m\left({\mathit{x}}_I\right)\right]$$

(6)

In the case of the RPIM, ${\mathit{R}}_{\mathit{i}\mathit{I}}\left({\mathit{x}}_I\right)$ is a multiquadric radial basis function (MQ-RBF) [41,44,51], whose form is generically expressed using the following equation:

$${\mathit{R}}_{\mathit{i}\mathit{I}}\left({\mathit{x}}_I\right)={(d_{Ii}^2+c^2)}^p$$

(7)

In this equation, $c$ and $p$ are defined as the shape parameters of the MQ-RBF. The efficiency of the method is dependent on them, but optimal values ($c=0.0001$ and $p=0.9999$) are presented in the literature [47]. Regarding $d_{Ii}$, also known as the Euclidean norm, it represents the distance between ${\mathit{x}}_I={\left\{x_i,y_i\right\}}^T$ and ${\mathit{x}}_I$. This can be defined as $d_{Ii}=\sqrt{{\left(x_i-x_I\right)}^2+{\left(y_i-y_I\right)}^2}$.

In the case of $\mathit{p}\left({\mathit{x}}_I\right)$, different forms of this function can be used. Nevertheless, a standard form of this function can be given by the next equation:

$${\mathit{p}\left(\mathit{x}\right)}^T=\left[1,x,y,x^2,xy,y^2,\dots \right]$$

(8)

During this procedure, and in order to obtain only one solution, the equation below needs to be considered and included [41,51].

$$\sum _{ i=1}^n{\mathit{p}}_j\left({\mathit{x}}_i\right)a_i\left({\mathit{x}}_i\right)=\mathbf{0}$$

(9)

Having said this and taking into account all the previous steps, Equation (2) can be written as:

$$\left\{\begin{array}{c}{\mathit{u}}_{\mathrm{s}}\\ \mathbf{0}\end{array}\right\}=\left[\begin{array}{cc}\mathit{R}& \mathit{P}\\ {\mathit{P}}^{\mathit{T}}& \mathbf{0}\end{array}\right]\left\{\begin{array}{c}\mathit{a}\\ \mathit{b}\end{array}\right\}=\mathit{G}\left\{\begin{array}{c}\mathit{a}\\ \mathit{b}\end{array}\right\}$$

(10)

In this, ${\mathit{u}}_{\mathrm{s}} \mathrm{i}\mathrm{s}$ the vector of the variable nodal values, and $\mathit{G},\mathit{R},$ and $\mathit{P}$ are the moment matrix, radial moment, and polynomial moment matrices, respectively. The last three matrices can be defined, respectively, by the following expressions:

$${\mathit{u}}_s={\left\{u_1,u_2,\dots u_n\right\}}^T$$

(11)

$$\mathit{R}=\left[\begin{array}{ccc}R\left(r_{11}\right)& R\left(r_{21}\right)& \begin{array}{cc}\dots & R\left(r_{n1}\right)\end{array}\\ R\left(r_{21}\right)& R\left(r_{22}\right)& \begin{array}{cc}\dots & R\left(r_{n2}\right)\end{array}\\ \begin{array}{c}⋮\\ R\left(r_{1n}\right)\end{array}& \begin{array}{c}⋮\\ R\left(r_{2n}\right)\end{array}& \begin{array}{c} ⋮\\ \begin{array}{cc}\dots & R\left(r_{nn}\right)\end{array}\end{array}\end{array}\right]$$

(12)

$$\mathit{P}=\left[\begin{array}{ccc}1& x_1& \begin{array}{cc}{y}_1& z_1\end{array}\\ 1& x_2& \begin{array}{cc}{y}_2& z_2\end{array}\\ \begin{array}{c}⋮\\ 1\end{array}& \begin{array}{c}⋮\\ x_n\end{array}& \begin{array}{cc}\begin{array}{c}⋮\\ y_n\end{array}& \begin{array}{c}⋮\\ z_n\end{array}\end{array}\end{array}\right]$$

(13)

Note that it is possible to obtain a unique solution for the non-constant coefficients $\mathit{a}$ and $\mathit{b}$ if the inverse of $\mathit{R}$ exists. In this case, these coefficients can be obtained through the next expression:

$$\left\{\begin{array}{c}\mathit{a}\\ \mathit{b}\end{array}\right\}={\mathit{G}}^{-1}\left\{\begin{array}{c}\mathit{u}\\ \mathbf{0}\end{array}\right\}$$

(14)

Thus, combining this expression with Equation (2), it can be redefined as:

$$\mathit{u}\left({\mathit{x}}_I\right)=\left\{{\mathit{R}}^{\mathrm{T}}\left({\mathit{x}}_I\right){\mathit{p}}^{\mathrm{T}}\left({\mathit{x}}_I\right)\right\}{\mathit{G}}^{-1}\left\{\begin{array}{c}\mathit{u}\\ \mathbf{0}\end{array}\right\}=\left\{\mathit{\phi }\left({\mathit{x}}_I\right),\mathit{\Psi }\left({\mathit{x}}_I\right)\right\}\left\{\begin{array}{c}\mathit{u}\\ \mathbf{0}\end{array}\right\}$$

(15)

where $\mathit{\phi }\left({\mathit{x}}_I\right)$ is defined as the interpolation function vector given by:

$$\mathit{\phi }\left({\mathit{x}}_I\right)=\left\{{\mathit{\phi }}_1\left({\mathit{x}}_I\right) {\mathit{\phi }}_2\left({\mathit{x}}_I\right) \dots {\mathit{\phi }}_{\mathit{n}}\left({\mathit{x}}_I\right)\right\}$$

(16)

and the residual vector that can be obtained by the expression below and it is given by ($\mathit{\Psi }\left({\mathit{x}}_I\right)$)

$$\mathit{\Psi }\left({\mathit{x}}_I\right)=\left\{{\mathit{\Psi }}_1\left({\mathit{x}}_I\right) {\mathit{\Psi }}_2\left({\mathit{x}}_I\right) \dots {\mathit{\Psi }}_{\mathit{m}}\left({\mathit{x}}_I\right)\right\}$$

(17)

Bearing this in mind, Equation (15) may be expressed as:

$$u\left({\mathit{x}}_I\right)=\mathbf{\Phi }{\left({\mathit{x}}_I\right)}^T\mathbf{u}=\left\{{\mathit{\phi }\left({\mathit{x}}_I\right)}^T,{\mathit{\Psi }\left({\mathit{x}}_I\right)}^T\right\}\left\{\begin{array}{c}\mathit{u}\\ \mathbf{0}\end{array}\right\}$$

(18)

and the partial derivatives of the interpolated field variable as:

$$\left\{\begin{array}{c}\frac{{\partial u}^h\left({\mathit{x}}_I\right)}{\partial x}=\frac{\partial \mathbf{\Phi }{\left({\mathit{x}}_I\right)}^T}{\partial x}\mathit{u}=\left\{\frac{{\partial \mathit{\phi }\left({\mathit{x}}_I\right)}^T}{\partial x}; \frac{{\partial \mathit{\Psi }\left({\mathit{x}}_I\right)}^T}{\partial x}\right\}\left\{\begin{array}{c}\mathit{u}\\ \mathbf{0}\end{array}\right\}\\ \frac{{\partial u}^h\left({\mathit{x}}_I\right)}{\partial y}=\frac{\partial \mathbf{\Phi }{\left({\mathit{x}}_I\right)}^T}{\partial y}\mathit{u}=\left\{\frac{{\partial \mathit{\phi }\left({\mathit{x}}_I\right)}^T}{\partial y}; \frac{{\partial \mathit{\Psi }\left({\mathit{x}}_I\right)}^T}{\partial y}\right\}\left\{\begin{array}{c}\mathit{u}\\ \mathbf{0}\end{array}\right\}\end{array}\right.$$

(19)

Assuming that matrix $\mathit{G}$ is independent on ${\mathit{x}}_I$, it is possible to reorganise the terms above in order to be defined as:

$$\left\{\begin{array}{c} \left\{\frac{{\partial \mathit{\phi }\left({\mathit{x}}_I\right)}^T}{\partial x}; \frac{{\partial \mathit{\Psi }\left({\mathit{x}}_I\right)}^T}{\partial x}\right\}=\frac{\partial \left(\left\{{\mathit{R}}^{\mathrm{T}}\left({\mathit{x}}_I\right);{\mathit{p}}^{\mathrm{T}}\left({\mathit{x}}_I\right)\right\}{\mathit{G}}^{-1}\right)}{\partial x}=\left\{\frac{\partial {\mathit{R}}^{\mathrm{T}}\left({\mathit{x}}_I\right)}{\partial x}; \frac{\partial {\mathit{p}}^{\mathrm{T}}\left({\mathit{x}}_I\right)}{\partial x}\right\}{\mathit{G}}^{-1}\\ \left\{\frac{{\partial \mathit{\phi }\left({\mathit{x}}_I\right)}^T}{\partial y}; \frac{{\partial \mathit{\Psi }\left({\mathit{x}}_I\right)}^T}{\partial y}\right\}=\frac{\partial \left(\left\{{\mathit{R}}^{\mathrm{T}}\left({\mathit{x}}_I\right);{\mathit{p}}^{\mathrm{T}}\left({\mathit{x}}_I\right)\right\}{\mathit{G}}^{-1}\right)}{\partial y}=\left\{\frac{\partial {\mathit{R}}^{\mathrm{T}}\left({\mathit{x}}_I\right)}{\partial y}; \frac{\partial {\mathit{p}}^{\mathrm{T}}\left({\mathit{x}}_I\right)}{\partial y}\right\}{\mathit{G}}^{-1}\end{array}\right.$$

(20)

Through this, the partial derivatives of $\mathit{r}\left({\mathit{x}}_{\mathit{I}}\right)$ might be obtained in terms of $\mathit{x}$ and $\mathit{y}$. To obtain this, the following expression is used:

$$\left\{\begin{array}{c}\frac{ {\partial r}_i\left({\mathit{x}}_I\right)}{\partial x}=2p \left(x_i-x_I\right){\left(d_{Ii}^2+c^2\right)}^{p-1}\\ \frac{ {\partial r}_i\left({\mathit{x}}_I\right)}{\partial y}=2p \left(y_i-y_I\right){\left(d_{Ii}^2+c^2\right)}^{p-1}\end{array}\right.$$

(21)

Note that the interpolation function vector does not depend on the influence domain [41,47].

Since the RPIM utilises interpolation functions, they possess the Kronecker delta property ${\delta }_{ij}$ and exhibit local compact support. As a result, the essential boundary conditions can be directly enforced in the stiffness matrix [47].

2.4. Elasticity Equations

The elasticity theory can be used to simulate cell proliferation. Thus, considering the domain $\mathrm{\Omega }$, the equilibrium equations shown next can be used to describe a linear elastostatic problem:

$$\left\{\begin{array}{c}\nabla \cdot \mathbf{\Lambda }+\mathit{b}=0 in \Omega \\ \mathbf{\Lambda } \mathit{n}=\overline{\mathit{t}} on {\Gamma }_t\\ \mathit{u}=\overline{\mathit{u}} on {\Gamma }_u\end{array}\right.$$

(22)

Here, $\nabla$ represents the gradient operator and $\mathit{u}$ the displacement field. $\mathit{b}$ is the body force per unit volume, $\Gamma$ the contour which presents the essential ${\Gamma }_{\mathrm{u}}$ and the natural boundaries ${\Gamma }_t$, and $\mathsf{\Lambda }$ the Cauchy stress tensor for a kinematically admissible $\mathit{u}$, $\mathit{b}$ the body force per unit volume and. In addition,$\Gamma \in \Omega :{\Gamma }_u\cup {\Gamma }_t=\Gamma \bigwedge {\Gamma }_u\cap {\Gamma }_t=\varnothing$, $\overline{\mathit{u}}$ is the prescribed displacement on ${\Gamma }_u$, $\overline{\mathit{t}}$ the traction on ${\Gamma }_t$ and $\mathit{n}$ the unit outward normal to $\Gamma$.

Considering the Galerkin formulation written using the Voigt notation, the weak form equation can be given by the following equation:

$$\delta L={\int }_{\mathrm{\Omega }}\delta {\mathit{\epsilon }}^{\mathrm{T}}\mathit{\sigma }d\mathrm{\Omega }-{\int }_{\mathrm{\Omega }}\delta {\mathit{u}}^{\mathrm{T}}\mathit{b}d\mathrm{\Omega }-{\int }_{{\Gamma }_t}\delta {\mathit{u}}^T\overline{\mathit{t}} d{\Gamma }_t=0$$

(23)

where $\mathit{\epsilon }$ is the strain vector and $\mathit{\sigma }$ the stress vector.

In the case of $\mathit{\epsilon }$, this parameter can be obtained by:

$$\mathit{\epsilon }=\mathit{L}\mathit{u}$$

(24)

where $\mathit{L}$ is the differential operator defined as:

$$\mathit{L}={\left[\begin{array}{ccc}\frac{\partial }{\partial x}& 0& \frac{\partial }{\partial y}\\ 0& \frac{\partial }{\partial y}& \frac{\partial }{\partial x}\end{array}\right]}^{\mathbf{T}}$$

(25)

If the Hooke Law is assumed, then $\mathit{\sigma }$ can be obtained by resorting to the previous equation. Thus, $\mathit{\sigma }$ can also be expressed as a function of $\mathit{u}$:

$$\mathit{\sigma }=\mathit{c}\mathit{\epsilon }=\mathit{c}\mathit{L}\mathit{u}$$

(26)

where $\mathit{c}$ represents the material constitutive matrix, which is represented as:

$$\mathit{c}={\left[\begin{array}{ccc}\frac{1-{\upsilon }^2}{E}& -\frac{\upsilon +{\upsilon }^2}{E}& 0\\ -\frac{\upsilon +{\upsilon }^2}{E}& \frac{1-{\upsilon }^2}{E}& 0\\ 0& 0& \frac{1}{G}\end{array}\right]}^{-1}$$

(27)

In this matrix, $E$ is the elasticity modulus, $\upsilon$ is the material Poisson coefficient, and $G$ is the distortion modulus ($G=E/(2+2\upsilon )$).

In the case of 2D problems, each node ${\mathit{x}}_i$ is defined by two degrees of freedom, which are defined at ${\mathit{u}}_i=\left\{u_i,v_i\right\}$, the reference axis.

To obtain the virtual displacement $\delta \mathit{u}\left({\mathit{x}}_I\right)=\delta {\mathit{u}}_I$ in an interesting point ${\mathit{x}}_i$, the interpolation function can be redefined as:

$$\delta {\mathit{u}}_I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left\{\begin{array}{c}{\mathbf{\Phi }}_{\mathit{I}}\\ {\mathbf{\Phi }}_{\mathit{I}}\end{array}\right\}\delta {\mathit{u}}_S=\left[\begin{array}{ccc}{\phi }_1\left({\mathit{x}}_I\right)& 0& \cdots \\ 0& {\phi }_1\left({\mathit{x}}_I\right)& \cdots \end{array}\begin{array}{cc}{ \phi }_n\left({\mathit{x}}_I\right)& 0\\ 0& {\phi }_n\left({\mathit{x}}_I\right)\end{array}\right]\delta {\mathit{u}}_S$$

(28)

This way, and assuming the Hooke law and the strain/displacement relation, ${\int }_{\mathrm{\Omega }}\delta {\mathit{\epsilon }}^{\mathrm{T}}\mathit{\sigma }d\mathrm{\Omega }$ is equivalent to:

$${\int }_{\mathrm{\Omega }}\delta {\mathit{\epsilon }}^{\mathrm{T}}\mathit{\sigma }d\mathrm{\Omega }={\int }_{\mathrm{\Omega }}\delta {\left(\mathit{L}\delta \mathit{u}\right)}^{\mathrm{T}}\mathit{c}\left(\mathit{L}\mathit{u}\right)d\mathrm{\Omega }={\int }_{\mathrm{\Omega }}\delta {\left(\mathit{L}{\mathit{H}}_I\delta {\mathit{u}}_S\right)}^{\mathrm{T}}\mathit{c}\left(\mathit{L}{\mathit{H}}_I{\mathit{u}}_S\right)d\mathrm{\Omega }={\int }_{\mathrm{\Omega }}\delta {\left(\mathit{L}{\mathit{B}}_I^{\mathrm{T}}\mathit{c}{\mathit{B}}_{\mathit{I}}{\mathit{u}}_S\right)}^{\mathrm{T}}d\mathrm{\Omega }=\delta {\mathit{u}}^{\mathbf{T}}{\int }_{\mathrm{\Omega }}{\mathit{B}}_I^{\mathrm{T}}\mathit{c}{\mathit{B}}_{\mathit{I}}d\mathrm{\Omega }\mathit{u}=\delta {\mathit{u}}^{\mathbf{T}}\mathit{K}\mathit{u}$$

(29)

In which ${\mathit{B}}_{\mathit{I}}$ is defined as the deformation matrix for the $n$ nodes of the influence cell of ${\mathit{x}}_I$ and given by the next expression.

$${\mathit{B}}_{\mathit{I}}=\left[\begin{array}{ccc}\frac{\partial {\phi }_1\left({\mathit{x}}_I\right)}{\partial x}& 0& \frac{\partial {\phi }_2\left({\mathit{x}}_I\right)}{\partial x}\\ 0& \frac{\partial {\phi }_1\left({\mathit{x}}_I\right)}{\partial y}& 0\\ \frac{\partial {\phi }_1\left({\mathit{x}}_I\right)}{\partial y}& \frac{\partial {\phi }_1\left({\mathit{x}}_I\right)}{\partial x}& \frac{\partial {\phi }_2\left({\mathit{x}}_I\right)}{\partial y} \end{array}\begin{array}{ccc}0& \cdots & \frac{\partial {\phi }_n\left({\mathit{x}}_I\right)}{\partial x}\\ \frac{\partial {\phi }_2\left({\mathit{x}}_I\right)}{\partial y}& \cdots & 0\\ \frac{\partial {\phi }_2\left({\mathit{x}}_I\right)}{\partial x}& \cdots & \frac{\partial {\phi }_n\left({\mathit{x}}_I\right)}{\partial y}\end{array} \begin{array}{c}\begin{array}{c}0\\ \end{array}\\ \frac{\partial {\phi }_n\left({\mathit{x}}_I\right)}{\partial y}\\ \frac{\partial {\phi }_n\left({\mathit{x}}_I\right)}{\partial x}\end{array}\right]$$

(30)

Moreover, $\mathit{K}$ is the global stiffness matrix, and $\mathit{u}$ is the displacement field vector.

A similar procedure can be performed for the other two terms. Following the same idea, these can be expressed, respectively, as:

$${\int }_{\mathrm{\Omega }}\delta {\mathit{u}}^{\mathrm{T}}\mathit{b}d\mathrm{\Omega }={\int }_{\mathrm{\Omega }}{\left({\mathit{H}}_I\delta {\mathit{u}}_S\right)}^{\mathrm{T}}\mathit{b}d\mathrm{\Omega }=\delta {\mathit{u}}^{\mathbf{T}}{\int }_{\mathrm{\Omega }}{\mathit{H}}_I^{\mathrm{T}}\mathit{b}d\mathrm{\Omega }=\delta {\mathit{u}}^{\mathbf{T}}{\mathit{f}}_b$$

(31)

And

$${\int }_{{\Gamma }_t}\delta {\mathit{u}}^T\overline{\mathit{t}} d{\Gamma }_t={\int }_{{\Gamma }_t}{\left({\mathit{H}}_I\delta {\mathit{u}}_S\right)}^{\mathrm{T}}\overline{\mathit{t}} d{\Gamma }_t={\delta \mathit{u}}^T{\int }_{{\Gamma }_t}{\mathit{H}}_I^{\mathrm{T}} \overline{\mathit{t}} d{\Gamma }_t=\delta {\mathit{u}}^{\mathbf{T}}{\mathit{f}}_t$$

(32)

In these equations, ${\mathit{f}}_b$ is the global body force and ${\mathit{f}}_t$ the global external force.

In the end, Equation (23) can be rewritten, and the following equation is obtained:

$$\delta {\mathit{u}}^{\mathbf{T}}\mathit{K}\mathit{u}-\delta {\mathit{u}}^{\mathbf{T}}{\mathit{f}}_b-\delta {\mathit{u}}^{\mathbf{T}}{\mathit{f}}_t=0$$

(33)

Through this, it is possible to define the next equation:

$$\mathit{K}\mathit{u}={\mathit{f}}_b+{\mathit{f}}_t$$

(34)

And, consequently, $\mathit{u}$ can be obtained as follows:

$${\mathit{K}}^{-1}\mathit{f}=\mathit{u}$$

(35)

After obtaining $\mathit{u}$, $\mathit{\epsilon }$ and $\mathit{\sigma }$ at every ${\mathit{x}}_I\in \mathrm{\Omega }$ might be given by the next expressions:

$$\mathit{\epsilon }\left({\mathit{x}}_I\right)=\mathit{L}\mathit{u}\left({\mathit{x}}_I\right)$$

(36)

$$\mathit{\sigma }\left({\mathit{x}}_I\right)=\mathit{c}\left({\mathit{x}}_I\right)\mathit{\epsilon }\left({\mathit{x}}_I\right)$$

(37)

2.5. Chemical Reaction-Diffusion Equations

Meshless methods can be employed to formulate and solve partial differential equations if a chemical reaction–diffusion problem is under study [52].

As mentioned earlier, the influence of oxygen and glucose on tumour cell proliferation is significant, given that the whole process relies on the cells’ uptake of these molecules from the extracellular environment. Therefore, in this study, to mimic this behaviour, the diffusion of these molecules in a homogeneous medium was considered [53].

Considering a 2D problem and the general Helmholtz equation, a problem involving a linear stable state field can be expressed as:

$$D_x\frac{{\partial }^{2}C}{\partial x^2}+D_y\frac{{\partial }^{2}C}{\partial y^2}-gC+Q=0$$

(38)

where $C$ represents the field variable, which was defined as the concentration, $D_x$ and $D_y$ the diffusion coefficient in $x$ and $y$, $g$ the matrix of chemical infusibility, and $Q$ the source of $C$. In addition, this equation was assumed for glucose and oxygen, and $g$ was defined as 0.

If the weighted residual approach is considered, it is possible to define the meshless system as:

$$\left[{\mathit{K}}_D+{\mathit{K}}_g\right]\mathbf{C}-{\mathit{f}}_q=0$$

(39)

In order to obtain the meshless system equations, ${\mathit{K}}_D$ is given by:

$${\mathit{K}}_D={\int }_A{\mathit{B}}_G^T\mathit{D}{\mathit{B}}_{G}dA$$

(40)

In which ${\mathit{B}}_G$ and $\mathit{D}$ are obtained by the next matrices:

$$\begin{gathered} {\mathit{B}}_G=\left[\begin{array}{c}\frac{\partial \phi }{\partial x}\\ \frac{\partial \phi }{\partial y}\end{array}\right]=\left[\begin{array}{ccc}\frac{\partial {\phi }_1}{\partial x}& \frac{\partial {\phi }_2}{\partial x}& \dots \frac{\partial {\phi }_n}{\partial x}\\ \frac{\partial {\phi }_1}{\partial y}& \frac{\partial {\phi }_2}{\partial y}& \dots \frac{\partial {\phi }_n}{\partial y}\end{array}\right] \\ \mathit{D}=\left[\begin{array}{cc}{D}_x& 0\\ 0& D_y\end{array}\right] \end{gathered}$$

(41)

${\mathit{K}}_g$ represented as:

$${\mathit{K}}_g={\int }_{A}g{\left\{\begin{array}{cccc}{\phi }_1& {\phi }_2& \dots & {\phi }_n\end{array}\right\}}^T\left\{\begin{array}{cccc}{\phi }_1& {\phi }_2& \dots & {\phi }_n\end{array}\right\}dA$$

(42)

and ${\mathit{f}}_q$, which is the flux of concentration, as:

$${\mathit{f}}_q={\int }_{A}Q{\left\{\begin{array}{cccc}{\phi }_1& {\phi }_2& \dots & {\phi }_n\end{array}\right\}}^{T}dA$$

(43)

After this, the final $\mathit{C}$ concentration distribution might be obtained if the solution of Equation (39) is found.

3. Cellular Proliferation Algorithm Description

3.1. Proliferation Algorithm

The algorithm used in this study was developed to simulate the proliferation of a generic cell under normal and abnormal conditions. This process is dependent on the concentrations of oxygen (${\mathrm{O}}_2$) and glucose (GL), as well as a growth phenomenological law that is based on documented observations that describe the volume of the cell over time. An illustration of the main steps performed by the algorithm is presented in Figure 2.

To initiate the algorithm, specific data need to be input (referred to as the “Input Data” block in Figure 2). This includes the size of the domain, divisions along both the horizontal ($Ox$) and vertical ($Oy$) axes, cell size, maximum iteration count, and properties of the cell and the extracellular matrix (such as diffusion coefficients and concentrations of ${\mathrm{O}}_2$ and GL). Once these initial parameters are defined, the algorithm creates a nodal mesh, assigns material properties to the cell and extracellular medium, and applies the boundary conditions. The essential boundary conditions (${\Gamma }_{u_l}$ and ${\Gamma }_{u_d}$) were assumed to be periodic (double mirror) for all the limits of the problem. The diffusion of the molecules was allowed to occur from ${\Gamma }_u^{(+)}$ to ${\Gamma }_u^{(-)}$, as well as in the reverse direction, as shown in Figure 3a.

Subsequently, the background integration mesh is constructed using square integration cells and a regular lattice. Each side of the square cells has a dimension equal to twice the average nodal distance. Within these integration cells, the integration points are placed using the Gauss–Legendre quadrature scheme. Additionally, different materials are defined in the domain to differentiate the cell from the medium based on their geometric positions.

Using radial searches, the influence domains are created. For each integration point, a specific number of closest nodes are identified. This information is then used to construct the shape functions (i.e., the radial point interpolation functions), which allow for the interpolation of values within the influence domains. At this stage, an iterative loop begins, progressing through a fictitious time step.

During the iterations, the concentrations of oxygen and glucose are enforced in the domain by solving the reaction–diffusion equations. This process enables the determination of the distribution of these concentrations at each iteration.

Once these steps are completed and normal concentration values are assumed in the domain, the cell initiates the proliferation process by imposing a volume variation. Assuming the present iteration as the time step and considering the medium to be under hydrostatic strain (${\epsilon }_{xx}{=\epsilon }_{yy}={\epsilon }_{zz}$), the corresponding strain required for this volume variation can be determined to be $\mathsf{\Delta }\mathrm{V}={\epsilon }_{xx}{\epsilon }_{yy}{\epsilon }_{zz}$ and ${\epsilon }_{xx}{=\epsilon }_{yy}={\epsilon }_{zz}=\mathsf{\Delta }\mathrm{V}/3$. On the other hand, assuming that $\mathit{\sigma }=\mathit{c}\mathit{\epsilon }$, the stress field $\mathit{\sigma }$ is determined, and it is possible to be imposed in the solid domain through an internal body force: $\nabla \mathsf{\Lambda }+\mathit{b}=0\Rightarrow \mathit{b}=-\nabla \mathsf{\Lambda }$. The imposition of this pressure variation enforces growth in the cell nodes, which, in turn, causes an increase in the volume of the cell. The region outside the cell maintains the same pressure. This implies that the nodes inside the cell either cause it to shrink, grow, or maintain the same volume, as no pressure is imposed by the medium. The displacement field is obtained by calculating the internal pressure generated by the volume variation. As a result, a new nodal configuration can be established by adding the obtained displacements to the initial coordinates of the nodes. This new configuration is used in the subsequent iteration. This iterative process continues until the programme reaches a stopping condition.

To obtain the volume variation, the fictitious volume that the cell should have in a particular iteration is calculated based on the concentrations of glucose and oxygen.

If the mean concentrations of oxygen and glucose in the domain are within normal values (i.e., a concentration within the interval $\left[8.932\times {10}^{-15};1.116\times {10}^{-15}\right] \mathrm{m}\mathrm{o}\mathrm{l}/{\mu \mathrm{m}}^2$ of oxygen [8,15,52,56] or within $\left[1.944\times {10}^{-14};3.500\times {10}^{-14}\right] \mathrm{m}\mathrm{o}\mathrm{l}/{\mu \mathrm{m}}^2$ of glucose [26,53,57], the cell grows according to the original proposed phenomenological law proposed in [46]. This law is given by the following expression:

$$V_f=V_i \cdot {\left[2\cdot \frac{e^{\frac{t_i}{t_d}}}{e}-\left(\frac{2}{e}-1\right)\cdot \left(1-\frac{t_i}{t_d}\right)\right]}^n\cdot \frac{1}{2^n}\cdot \frac{t_i}{t_d}+V_i$$

(44)

where $V_f$ is defined as the volume in an iteration $i$, $V_i$ the initial volume, $t_d$ the time that is necessary for the cell to divide into two new cells, $t_i$ the time of each iteration, and $n$ specifies the inclination of the curve. This phenomenological law was obtained by curve-fitting experimental data available in the literature (Figure 3b) [54,55], and $n=2$ was used for the simulations as it was the value that resulted in the smallest difference among the experimental and theoretical curves, as demonstrated in [46].

In the case of normal concentrations, the cell continues to grow until it reaches twice its initial volume, at which point it undergoes division into two new cells. When the cell reaches double its volume, the “Division Block” is initiated (Figure 4) in order to divide the cell (Figure 5a). To do that, the process of growth is blocked, and two points ($X_A$ and $X_B$) are arbitrarily selected in opposite positions of the cell. After that, the points move to the midpoint of the cell until they are in the same position, having the same coordinates. In this case, the $∆\mathrm{V}$ inside the cell is 0, so it is possible to assume a hydrostatic stress state. This allows for the conversion of equivalent stress into equivalent forces, which can be used to determine the forces applied to the cell and enforce the displacement using the equation ${\mathit{K}}^{-1}·\mathit{f}=\mathit{u}$. That way, the cell is compressed until the two points are positioned at the centre of the cell, and then the division into two new cells takes place. When the process of division is completed, the algorithm stops.

When abnormal concentrations are present in the domain, they can affect the growth process, leading to four main implications: slower or faster growth velocity, maintenance of the same cell volume, or initiation of the apoptosis process. These modifications in the normal proliferation process depend on the combination of oxygen and glucose concentration states within the specific iteration of the domain. In addition to the normal concentration states of oxygen and glucose, there are several possible states: extreme hyperoxia, hyperoxia, hypoxia, and extreme hypoxia for oxygen concentration [15,52,58,59], and extreme hyperglycaemia, hyperglycaemia, hypoglycaemia, and extreme hypoglycaemia for glucose concentration [26,60,61]. The concentration ranges defining each case are presented in

Table 1. Range of concentrations that defined each oxygen (${\mathbf{O}}_2$) and glucose (GL) abnormal state used in the proposed algorithm.

${\mathbf{O}}_2$ State	$\left[{\mathbf{O}}_2\right]$$\mathbf{(}\mathbf{m}\mathbf{o}\mathbf{l}/{\mathbf{\mu }\mathbf{m}}^2$)	GL State	$\left[\mathbf{G}\mathbf{L}\right]$$\mathbf{(}\mathbf{m}\mathbf{o}\mathbf{l}\mathbf{/}{\mathbf{\mu }\mathbf{m}}^2$)
Extreme Hyperoxia	$>4.466\times {10}^{-14}$	Extreme Hyperglycaemia	$>7.5\times {10}^{-14}$
Hyperoxia	$\left]8.932\times {10}^{-15},4.466\times {10}^{-14}\right]$	Hyperglycaemia	$\left]3.5\times {10}^{-14},7.5\times {10}^{-14}\right]$
Hypoxia	$\left[2.233\times {10}^{-17},1.116\times {10}^{-15}\right[$	Hypoglycaemia	$\left[1.111\times {10}^{-14},1.944\times {10}^{-14}\right[$
Extreme Hypoxia	$<2.233\times {10}^{-17}$	Extreme Hypoglycaemia	$<1.111\times {10}^{-14}$

.

For each case, the phenomenological law was modified to achieve the desired growth behaviour. These modifications are detailed in the following section.

3.2. Influence of Oxygen and Glucose Concentrations

The different combinations of concentration states have a significant impact on the growth of the cell over time. When hyperoxia, hypoxia, and/or hypoglycaemia have been determined, the growth rate will be slower compared to when normal oxygen and/or glucose concentrations are present, since these three conditions tend to decrease the rate of cellular proliferation for both types of cells.

For hyperoxia ($a_{O_2}$) and hypoxia ($b_{O_2}$), a parameter α was introduced to influence the growth of cell volume in a specific iteration while considering the specific concentration range for each case. The value of α was determined through the following expressions:

$$\begin{gathered} {\mathsf{\alpha }}_{a_{O_2}m}=-2.79899\times {10}^{13}{C}_{O_{2}m}+1.25 \\ {C}_{O_{2}m} \in [8.932\times {10}^{-15},4.466\times {10}^{-14}]\mathrm{m}\mathrm{o}\mathrm{l}/{\mu \mathrm{m}}^2 \\ {\mathsf{\alpha }}_{b_{O_2}m}=9.13954\times {10}^{14}{C}_{O_{2}m}-2.04082\times {10}^{-2} \\ {C}_{O_{2}m} \in [2.233\times {10}^{-17},1.116\times {10}^{-15}]\mathrm{m}\mathrm{o}\mathrm{l}/{\mu \mathrm{m}}^2 \end{gathered}$$

(45)

In these, $C_{O_2}$ represents the mean oxygen concentration of the domain in a specific iteration $m$.

Regarding the healthy cell, a similar process was assumed for hyperglycaemia ($a_G$) and hypoglycaemia ($b_G)$. For these conditions, in order to affect the growth of the cell, a parameter $\mathsf{\beta }$ was introduced to slow the process of growth. The parameter $\mathsf{\beta }$ is obtained for these cases as follows:

$$\begin{gathered} {\mathsf{\beta }}_{a_G}=-2.5\times {10}^{13}{C}_{Gm}-1.875 \\ {C}_G \in [3.5\times {10}^{-14},7.5\times {10}^{-14}] \mathrm{m}\mathrm{o}\mathrm{l}/{\mu \mathrm{m}}^2 \\ {\mathsf{\beta }}_{b_G}=1.2\times {10}^{14}{C}_{Gm}-1.333 \\ {C}_G \in [1.111\times {10}^{-14},1.944\times {10}^{-14}]\mathrm{m}\mathrm{o}\mathrm{l}/{\mu \mathrm{m}}^2 \end{gathered}$$

(46)

Here, $C_G$ is the mean glucose concentration of the domain in the iteration $m$.

It should be noted that when these four states are combined with each other or with normal concentrations of oxygen or glucose, the more aggressive state dominates. As a result, different combinations also result in variable durations of the proliferation process.

For tumour cells under hypoglycaemia, the procedure defined for healthy cells remains unchanged. However, in the case of hyperglycaemia ($a_G$), a new equation was introduced. Therefore, if the oxygen concentration is normal, the proliferation occurs at a faster rate compared to the normal glucose concentration. Hence, for hyperglycaemia, the parameter $\mathsf{\beta }$ is determined by the following equation:

$$\begin{gathered} {\mathsf{\beta }}_{a_{G}m}=2.5\times {10}^{13}{C}_{Gm}-0.125 \\ {C}_{Gm} \in [3.5\times {10}^{-14},7.5\times {10}^{-14}] \mathrm{m}\mathrm{o}\mathrm{l}/{\mu \mathrm{m}}^2 \end{gathered}$$

(47)

However, if this state is combined with hyperoxia or hypoxia, it is assumed that growth is negatively affected. Nevertheless, it is also assumed that the process is not as slow as when these states are combined with a normal glucose concentration or with hypoglycaemia. To achieve this behaviour, specific $\mathsf{\alpha }$ values were obtained for these two particular combinations (hyperoxia/hyperglycaemia and hypoxia/hyperglycaemia, respectively):

$$\begin{gathered} {\mathsf{\alpha }}_{a_{O_2}m}=-1.39949\times {10}^{13}{C}_{O_{2}m}+1.125 \\ {C}_{O_{2}m} \in [8.932\times {10}^{-15},4.466\times {10}^{-14}]\mathrm{m}\mathrm{o}\mathrm{l}/{\mu \mathrm{m}}^2 \\ {\mathsf{\alpha }}_{b_{O_2}m}=4.56977\times {10}^{14}{C}_{O_{2}m}-0.4898 \\ {C}_{O_{2}m} \in [2.233\times {10}^{-17},1.116\times {10}^{-15}]\mathrm{m}\mathrm{o}\mathrm{l}/{\mu \mathrm{m}}^2 \end{gathered}$$

(48)

Regarding the extreme states (extreme hyperoxia ($c_{O_2}$), extreme hypoxia ($d_{O_2}$), extreme hyperglycaemia ($e_G$) and extreme hypoglycaemia ($f_G$)), cells are highly influenced by these changes in oxygen and glucose concentrations. Under these states, cells tend to cease growing and undergo cell death instead. Therefore, in the algorithm, cells were not allowed to grow for a specific period of time, and if this state was not reversed, the cell would initiate the process of apoptosis (Figure 5b). To define this last step, the parameters α and $\mathsf{\beta }$ were defined as follows:

$$\begin{gathered} if t_m>t_{O_2} \\ {\alpha }_{{c/d}_{O_2}}=\frac{1}{t_m}\cdot \left(t_{O_2}-t_m\right) \\ if t_m>t_G \\ {\beta }_{{e/f}_G}=\frac{1}{t_m}\cdot \left(t_G-t_m\right) \end{gathered}$$

(49)

where ${\mathrm{t}}_{\mathrm{m}}$ represents the time of the iteration $m$ in which the algorithm is, and $t_{O_2}$ and $t_G$ represent the durations for which the cell needs to remain in extreme oxygen or glucose conditions, respectively, in order to initiate the process of apoptosis. For extreme hyperoxia and extreme hypoxia, $t_{O_2}$ was assumed to be 36 h if the concentration of glucose was within its normal range. In the case of extreme hyperglycaemia or extreme hypoglycaemia, the variable ${\mathrm{t}}_{\mathrm{G}}$ was assumed to be 24 h when $C_{O_2}$ is within its normal range. Note that these α and $\mathsf{\beta }$ parameters only affect the growth process when the time periods mentioned above have passed ($t_m\ge t_{O_2}$ or $t_m\ge t_G$), triggering the process of apoptosis, which leads to a 40% reduction in the initial cell volume and ultimately results in cell death [62,63,64,65]. Until then ($t_m<t_{O_2}\mathrm{o}\mathrm{r} t_m<t_G$), the cell volume remains constant if the extreme state in which the cell is located does not change. When extreme cases are combined with another abnormal state, the variables $t_{O_2}$ and $t_G$ are defined with lower values, depending on the aggressiveness of the other combined state. In cases where two extreme states are combined, the values of $t_{O_2}$ and $t_G$ are assumed to be equal. Thus, a generic $t_e$ was considered, and a similar expression to the ones presented above was taken into account:

$$\begin{gathered} if t_m>t_e \\ {\alpha }_e=\frac{1}{t_m}\cdot \left(t_e-t_m\right) \end{gathered}$$

(50)

Table 2. Schematic chart of the conditions implemented in the algorithm for the twenty-five combinations when a healthy cell is assumed.

	Extreme Hypoxia	Hypoxia	Normal Level Oxygen	Hyperoxia	Extreme Hyperoxia
Extreme Hyperglycemia	Constant Volume + Apoptosis ${\mathrm{t}}_{\mathrm{e}}=12$	Constant Volume + Apoptosis ${\mathrm{t}}_{\mathrm{G}}=24$	Constant Volume + Apoptosis ${\mathrm{t}}_{\mathrm{G}}=36$	Constant Volume + Apoptosis ${\mathrm{t}}_{\mathrm{G}}=30$	Constant Volume + Apoptosis ${\mathrm{t}}_{\mathrm{e}}=18$
Hyperglycemia	Constant Volume + Apoptosis ${\mathrm{t}}_{{\mathrm{O}}_2}=16$	Slower Growth ${\mathsf{\alpha }}_{b_{O_2}m}\in \left[\mathrm{0,1}\right]$ ${\mathsf{\beta }}_{a_{G}m}\in \left[\mathrm{0,1}\right]$	Slower Growth ${\mathsf{\beta }}_{a_{O_2}m}\in \left[\mathrm{0,1}\right]$	Slower Growth ${\mathsf{\alpha }}_{a_{O_2}m}\in \left[\mathrm{0,1}\right]$ ${\mathsf{\beta }}_{a_{G}m}\in \left[\mathrm{0,1}\right]$	Constant Volume + Apoptosis ${\mathrm{t}}_{{\mathrm{O}}_2}=22$
Normal Level Glucose	Constant Volume + Apoptosis ${\mathrm{t}}_{{\mathrm{O}}_2}=24$	Slower Growth ${\mathsf{\alpha }}_{b_{O_2}m}\in \left[\mathrm{0,1}\right]$	Normal Growth ${\mathsf{\alpha }}_m=1$ ${\mathsf{\beta }}_m=1$	Slower Growth ${\mathsf{\alpha }}_{a_{O_2}m}\in \left[\mathrm{0,1}\right]$	Constant Volume + Apoptosis ${\mathrm{t}}_{{\mathrm{O}}_2}=24$
Hypoglycemia	Constant Volume + Apoptosis ${\mathrm{t}}_{{\mathrm{O}}_2}=8$	Slower Growth ${\mathsf{\alpha }}_{b_{O_2}m}\in \left[\mathrm{0,1}\right]$ ${\mathsf{\beta }}_{b_{G}m}\in \left[\mathrm{0,1}\right]$	Slower Growth ${\mathsf{\beta }}_{b_{O_2}m}\in \left[\mathrm{0,1}\right]$	Slower Growth ${\mathsf{\alpha }}_{b_{O_2}m}\in \left[\mathrm{0,1}\right]$ ${\mathsf{\beta }}_{a_{G}m}\in \left[\mathrm{0,1}\right]$	Constant Volume + Apoptosis ${\mathrm{t}}_{{\mathrm{O}}_2}=20$
Extreme Hypoglycemia	Constant Volume + Apoptosis ${\mathrm{t}}_{\mathrm{e}}=4$	Constant Volume + Apoptosis ${\mathrm{t}}_{\mathrm{G}}=18$	Constant Volume + Apoptosis ${\mathrm{t}}_{\mathrm{G}}=36$	Constant Volume + Apoptosis ${\mathrm{t}}_{\mathrm{G}}=24$	Constant Volume + Apoptosis ${\mathrm{t}}_{\mathrm{e}}=16$

and

Table 3. Schematic chart of the conditions implement in the algorithm for the twenty-five combinations, when a tumour cell is assumed.

	Extreme Hypoxia	Hypoxia	Normal Level Oxygen	Hyperoxia	Extreme Hyperoxia
Extreme Hyperglycemia	Constant Volume + Apoptosis ${\mathrm{t}}_e=12$	Constant Volume + Apoptosis ${\mathrm{t}}_{\mathrm{G}}=24$	Constant Volume + Apoptosis ${\mathrm{t}}_{\mathrm{G}}=36$	Constant Volume + Apoptosis ${\mathrm{t}}_{\mathrm{G}}=30$	Constant Volume + Apoptosis ${\mathrm{t}}_e=18$
Hyperglycemia	Constant Volume + Apoptosis ${\mathrm{t}}_{{\mathrm{O}}_2}=16$	Slower Growth ${\mathsf{\alpha }}_{b_{O_2}m}\in \left[\mathrm{0.5,1}\right]$ ${\mathsf{\beta }}_{a_{G}m}\in \left[\mathrm{0,1}\right]$	Faster Growth ${\mathsf{\beta }}_{a_{G}m}\in \left[\mathrm{1,2}\right]$	Slower Growth ${\mathsf{\alpha }}_{a_{O_2}m}\in \left[\mathrm{0.5,1}\right]$ ${\mathsf{\beta }}_{a_{G}m}\in \left[\mathrm{0,1}\right]$	Constant Volume + Apoptosis ${\mathrm{t}}_{{\mathrm{O}}_2}=22$
Normal Level Glucose	Constant Volume + Apoptosis ${\mathrm{t}}_{{\mathrm{O}}_2}=24$	Slower Growth ${\mathsf{\beta }}_{a_{O_2}m}\in \left[\mathrm{0,1}\right]$	Normal Growth ${\mathsf{\alpha }}_m=1$ ${\mathsf{\beta }}_m=1$	Slower Growth ${\mathsf{\beta }}_{a_{O_2}m}\in \left[\mathrm{0,1}\right]$	Constant Volume + Apoptosis ${\mathrm{t}}_{{\mathrm{O}}_2}=24$
Hypoglycemia	Constant Volume + Apoptosis ${\mathrm{t}}_{{\mathrm{O}}_2}=8$	Slower Growth ${\mathsf{\alpha }}_{b_{O_2}m}\in \left[\mathrm{0,1}\right]$ ${\mathsf{\beta }}_{b_{G}m}\in \left[\mathrm{0,1}\right]$	Slower Growth ${\mathsf{\beta }}_{b_{G}m}\in \left[\mathrm{0,1}\right]$	Slower Growth ${\mathsf{\alpha }}_{a_{O_2}m}\in \left[\mathrm{0,1}\right]$ ${\mathsf{\beta }}_{b_{G}m}\in \left[\mathrm{0,1}\right]$	Constant Volume + Apoptosis ${\mathrm{t}}_{{\mathrm{O}}_2}=20$
Extreme Hypoglycemia	Constant Volume + Apoptosis ${\mathrm{t}}_e=4$	Constant Volume + Apoptosis ${\mathrm{t}}_{\mathrm{G}}=18$	Constant Volume + Apoptosis ${\mathrm{t}}_{\mathrm{G}}=36$	Constant Volume + Apoptosis ${\mathrm{t}}_{\mathrm{G}}=24$	Constant Volume + Apoptosis ${\mathrm{t}}_e=16$

summarise what occurred in each state for the healthy and tumour cells, respectively.

4. Results and Discussion

The evolution of the volume of a healthy and a tumour cell over time was analysed in order to study the influence of oxygen and glucose on cell growth and, at the same time, the viability of the algorithm to simulate the twenty-five proposed combinations. As a reference, the duration of a cell cycle (i.e., from the beginning of the growth process until two cells are obtained), under normal conditions of oxygen and glucose, was defined as 24 h (0.857 h per iteration), for a healthy cell, and as 18 h (0.643 h per iteration), for a tumour cell [66,67,68]. Regarding the initial parameters that were used, the size of the domain was set to $1000 per 1000 \mu \mathrm{m}$, and the extracellular part was assumed to be homogenised with an elasticity modulus of 1 Pa [69] and a Poisson’s coefficient of 0.45 [69,70,71]. The width of the cell was set to $100 \mu \mathrm{m}$ [72,73], and the diffusion coefficients were defined as 2000 $\mu \mathrm{m}/\mathrm{s}$ (cell) and 3030 $\mu \mathrm{m}/\mathrm{s}$ (medium) for oxygen and as 2000 $\mu \mathrm{m}/\mathrm{s}$ (cell) and 3030 $\mu \mathrm{m}/\mathrm{s}$ (medium) for glucose [7,23]. In addition, the same arrangement of the nodes and the same integration structure were used. A representation of these meshes can be seen in Figure 6. For all analyses, an integration scheme of $6\times 6$ integration points in each integration cell and influence domains with seven nodes were assumed. These parameters were optimised in a preliminary numerical study to increase the efficiency of the code (reduce the computational cost and maintain the accuracy and stability of the solution) and remained constant in all analyses. Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16 present the growth evolution of healthy and tumour cells over time and for each combination.

In Figure 7 (healthy cell) and Figure 8 (tumour cell), the concentration of glucose was assumed to be within the extreme hyperglycaemia range, while the concentration of oxygen was varied among the five possible ranges. For all curves and both types of cells, the cell size remained constant for a period of time (as indicated in Table 2 and Table 3). After this, the cell initiated the process of apoptosis, resulting in a reduction of the cell volume by up to 40% of its initial volume. This behaviour can be attributed to the extreme hyperglycaemia condition, which was present in all cases and defined in the algorithm as the only possible outcome. However, it is important to note that the duration of the constant volume period and the initiation of apoptosis varied for each combination. More aggressive combinations, such as extreme hyperglycaemia combined with extreme hypoxia, resulted in shorter analysis times, leading to earlier cell death. The combination with normal concentrations of oxygen took 24 h to initiate the apoptosis process, as defined in the algorithm.

Figure 9 illustrates the volume variation of the healthy cell over time for different oxygen concentration states, considering hyperglycaemia. Once again, when extreme conditions of oxygen are considered, the cell presents a similar behaviour as the one presented in Figure 7. In the remaining three cases, the growth of the cell is slower, as expected, since in each one there is at least one case of an abnormal concentration of glucose and oxygen. These conditions always have a negative impact on the proliferation process. However, in these three cases, the cell managed to grow and double its initial volume before undergoing the process of division. The tumour cell (Figure 10) analysed under the same combinations once again led to the cell maintaining a constant volume for a period of time before entering apoptosis. When hyperglycaemia and normal levels of oxygen are combined, the process of proliferation is enhanced, so the cell achieves double its volume in less than 18 h (the defined time for a tumour cell to complete the cell cycle). This acceleration can be attributed to the fact that hyperglycaemia promotes tumour cell proliferation. In combination with hyperoxia or hypoxia, as defined in the algorithm, the cell is allowed to grow and double its volume, but at a slower rate. However, since these states are combined with hyperglycaemia, their aggressiveness is mitigated.

Figure 11 and Figure 12 focused on analysing the effects of different oxygen states while maintaining a normal concentration of glucose. Similar to previous examples, extreme oxygen concentrations resulted in a similar and consistent pattern. However, since these extreme concentrations of oxygen were combined with a normal concentration of glucose, the cell remained at a constant volume for 36 h before entering apoptosis, as defined in the algorithm. When normal concentrations of glucose and oxygen were combined, the process of proliferation occurred within 24 h for the healthy cell and in 18 h for the tumour cell, which corresponds to a normal cell cycle. Comparing these results with the combination involving hyperoxia and hypoxia, it is evident that these two conditions significantly slowed down the proliferation process. However, in the case of the healthy cell, when comparing these combinations to the corresponding curves in Figure 9, where the concentration of glucose was defined in the hyperglycaemia range, it becomes apparent that more aggressive combinations lead to slower proliferation processes. On the other hand, for the tumour cell, combining normal glucose levels with hyperoxia and hypoxia resulted in slower growth compared to the corresponding curves in Figure 10. This outcome is expected since the domain had lower glucose values in these cases.

In Figure 13 and Figure 14, a hypoglycaemia state was considered, and similar curves are presented. The results closely resemble those obtained in Figure 9, which is expected since the combinations are approximately the same. In extreme cases, the cell begins to undergo apoptosis after a certain period of time. When hypoglycaemia is combined with the three other oxygen states, the process varies in speed depending on the aggressiveness of the environment. However, the duration of the process is always longer than 18 and 24 h, which are the durations for tumour and healthy cell cycles, respectively, under normal conditions of glucose and oxygen.

Lastly, in Figure 15 and Figure 16, extreme hypoglycaemia is combined with the five states of oxygen. Since extreme hypoglycaemia represents an extreme condition, all curves show a pause in the proliferation process (i.e., the cell maintains its initial volume) for a certain period of time, followed by the initiation of apoptosis, resulting in a reduction of 40% of the initial volume. However, as the combinations become more favourable, such as extreme hypoglycaemia combined with a normal concentration of oxygen, the cell survives for a longer period of time.

5. Conclusions

Cancer is one of the deadliest diseases in our society, affecting and killing millions of people every year. It is often associated with the emergence of mutations that promote abnormal proliferation of the affected cells, leading to an imbalance between cell proliferation and cell death. Due to that, the normal functioning of the tissues can be disrupted. To understand this problem, it is necessary to comprehend the mechanism and the species that are involved in the process of proliferation, both in healthy and diseased conditions. In this particular process, cells are strongly dependent on oxygen, nutrients, and growth factors. Therefore, understanding their influence becomes essential.

Various methods have been employed to study cancer, including mathematical and computational models, which have emerged as powerful tools to complement more traditional analyses. In the literature, different types of models can be found, and several numerical methods are used to solve them. Among these numerical methods, meshless methods, such as the RPIM, have emerged as an alternative to conventional methods and are currently employed to address more complex problems. However, when it comes to studying cell proliferation and cancer, these methods are less commonly utilized. One of the main reasons to choose this method is that mesh-based methods would require a more complex remeshing procedure since the elements of the mesh would have to be divided to simulate the process of cell division. This reduces the efficiency of the method. Thus, for the study of cell proliferation, which is a nonlinear geometric problem that implies remeshing, meshless methods are a better choice. In contrast to other meshless techniques, the RPIM is able to use structures available in FEM softwares and can perform structural analyses that are not possible with discrete methods that use probabilities and statistical laws, in contrast to other meshless techniques.

The objective of the present work was to implement and study the influence of oxygen and glucose concentrations on healthy and tumour cells using a previously created algorithm by the research group. The algorithm simulates the growth and division process of a single cell, which is dependent on oxygen and glucose.

To incorporate the influence of these two molecules, five states of concentration were defined for each one (extreme hyperoxia, hyperoxia, normal oxygen concentration, hypoxia, and extreme hypoxia for oxygen, extreme hyperglycaemia, hyperglycaemia, normal glucose concentration, hypoglycaemia, and extreme hypoglycaemia for glucose). The growth of the cell over time was analysed for each combination of oxygen and glucose states. In the end, it was possible to verify that the obtained results were coherent with what was expected, and each case led to satisfactory and adequate results for each one of the twenty-five combinations in terms of volume growth for both types of cells.

When normal concentrations were tested, the proliferation process occurred within 24 h for the healthy cell and within 18 h for the tumour cell, which aligns with the normal duration of a human cell cycle. However, when these concentrations were increased or decreased, the process of proliferation was affected, usually resulting in slower volume growth. This behaviour was observed in conditions of hyperoxia, hypoxia, and hypoglycaemia. When extreme states were analysed, the cell maintained the same volume until the initiation of the apoptosis process. Such behaviour is coherent with what occurs in the biological environment since, in extreme cases, cells tend to stop this process in order to protect themselves from the hostile environment and prevent the emergence and spread of mutations. In the case of a healthy cell, under hyperglycaemia, the same behaviour was verified. However, when the tumour cell is subjected to this state, the opposite effect is verified, leading to faster cell growth. Hyperglycaemia tends to favour the proliferation of tumour cells, highlighting one of the main differences between tumour cells and healthy cells. In summary, the main differences between healthy and tumour cells lie in the proliferation time and the impact of hyperglycaemia. In tumour cells, the double of the initial cell volume is achieved earlier, and in healthy cells, the state of hyperglycaemia negatively affects the growth of the healthy cell. Both behaviours were successfully implemented and simulated using the proposed algorithm.

Furthermore, the implementation of the RPIM as a discretization technique yielded accurate and satisfactory results. However, it should be noted that alternative methods can be considered. The literature offers various discretization techniques that may potentially enhance efficiency. Therefore, this research topic holds potential for further exploration.

Note that the proliferation of tumour cells is dependent on a huge number of parameters that can influence their progression and aggressiveness, often involving complex mechanisms. Therefore, considering only the concentrations of oxygen and glucose as factors influencing cell growth is a simplified approach.

Future work should focus on several improvements to enhance the algorithm. For instance, the inclusion of a larger number of cells in the domain would enable the study of cell populations and their interactions. The process of angiogenesis, which strongly influences tumour cells, should also be incorporated. Additionally, other parameters affecting tumour cell proliferation, such as pH levels, should be taken into account. These could open up new potential applications for our research, such as predicting the cellular behaviour of distinct cell types and the dynamics of various cell populations by applying this methodology with tailored parameters. It may also be applied to model the progression of diseases characterised by abnormal cell proliferation to obtain insights into disease dynamics and potential intervention points, enhancing our understanding of cellular behaviours in pathological contexts. Moreover, it can be considered for applications in a broader range of scenarios, from optimising cell culture conditions in laboratories to predicting responses in complex biological systems.