ϑ-Fractional Stochastic Models for Simulating Tumor Growth and Chemical Diffusion in Biological Tissues

Abstract

This paper presents an advanced simulation-based investigation of tumor growth and chemical diffusion in biological tissues, using $\vartheta$-fractional stochastic integral equations. Based on the theoretical framework developed in [Fractal Fract. 2025, 9(1), 7], we develop an innovative computational model to explore the practical applications of these equations in the biological field. The model focuses on providing new insights into the dynamic interaction between stochastic effects of a fractional nature and complex biological tissue environments, contributing to a deeper understanding of the mechanisms of chemical diffusion within tissues and tumor growth under different conditions. The paper details the numerical techniques used to solve the $\vartheta$-fractional stochastic integral equations, focusing on the stability and accuracy of the solutions, while demonstrating their ability to accurately and effectively capture key biological phenomena. Through extensive computational experiments, the model demonstrates its ability to replicate realistic tumor growth patterns and complex chemical transport dynamics, providing a powerful and flexible tool for understanding tumor behavior and interaction with potential therapies. These results represent an important step toward improving biological models and enhancing biomedical applications, particularly in the areas of targeted drug design and analysis of tumor dynamics under chemotherapeutic influence.

1. Introduction

Fractional calculus, which deals with fractional-order differentiation and integration, has emerged as a pivotal mathematical framework for modeling and analyzing complex systems across various scientific and engineering disciplines. This advanced branch of mathematics goes beyond the traditional limitations of integer-order calculus, providing precise tools for understanding nonlinear, nonlocal, and memory-dependent phenomena that are a fundamental feature of many natural and artificial systems. Applications of fractional calculus extend to diverse fields including electrical engineering, where it is used to improve the efficiency of power systems and filter design, control theory, with a focus on improving the stability of dynamical systems, thermal systems, through the analysis of complex heat conduction, signal processing, where it contributes to the analysis of nonlinear and time-dependent signals, and biological systems, where it is employed to model history-dependent physiological processes such as the transport of biomaterials or cell dynamics. Fractional calculus enables scientists and engineers to address multiple research and application challenges such as asymptotic stability analysis [1,2], which studies the long-term stability of systems, and finite-time stability [3,4], which focuses on the stability of systems over a specific time. It also plays a fundamental role in controller design, which helps improve the control of dynamical systems, and error estimation, which enhances mathematical models’ accuracy in predicting complex systems’ behavior. With these unique capabilities, fractional calculus provides a comprehensive and flexible framework for modeling multidisciplinary systems, making it an indispensable tool in addressing modern scientific and engineering challenges.

The introduction of fractional $\vartheta$ derivatives ($\vartheta$-FDs) and fractional $\vartheta$ integrals ($\vartheta$-FIs) is a notable development in fractional calculus, providing additional flexibility in describing systems exhibiting complex diffusion dynamics including subdiffusion and superdiffusion [5]. These tools can reconcile the transitional properties of dynamical systems, allowing for more accurate modeling of phenomena combining slow (constrained) and fast (unconstrained) diffusion in various real applications. Recent studies have investigated the theoretical properties of equations involving $\vartheta$-FDs and $\vartheta$-FIs, focusing on crucial aspects such as the existence, uniqueness, and stability [6,7]. Furthermore, the researchers in [5] proposed an innovative $\vartheta$-subdiffusion equation formulation, specifically designed to model complex transitions between classical subdiffusion and superdiffusion. This application trend reflects the increasing importance of $\vartheta$-fractional derivatives and integrals in addressing the challenges of mathematical modeling of nonlinear and memory-dependent phenomena, such as particle transport in porous media or chemical diffusion in biological tissues. This development highlights the profound potential of $\vartheta$-FDs and $\vartheta$-FIs in achieving significant advances in both theory and application, enhancing the status of fractional calculus as an essential mathematical tool in the study of complex systems.

In the field of stochastic processes, fractional stochastic integral equations (FSIEs) have become a vital mathematical tool due to their unique ability to incorporate the effects of randomness and long memory into mathematical models. These equations provide a flexible framework for describing systems that are affected by stochastic factors and exhibit memory-dependent dynamics, making them of particular interest in various fields such as statistical physics, biological systems, and financial markets. One of the fundamental concepts studied in this context is Ulam–Hayers stability (UHS), which addresses the sensitivity of solutions to perturbations in initial conditions or parameters. This type of stability is of particular importance in practical applications, where models are susceptible to small perturbations due to experimental errors or numerical approximations. UHS has been extensively studied for various types of fractional stochastic equations, with studies [8,9,10] showing that robust stability can be achieved even in systems with complex dynamical behavior. However, research on the Ulam–Hayer stability of $\vartheta$-FSIEs remains limited. This class of equations is a natural and important extension of conventional FSIEs, offering an additional degree of flexibility in describing transitional phenomena between subdiffusion and superdiffusion. The investigation of UHS for $\vartheta$-FSIEs represents an important step towards improving our understanding of complex multi-effect systems and holds broad potential for practical applications in modeling stochastic phenomena with memory-dependent dynamics, such as chemical diffusion in biological tissues or particle motion in random media.

The theoretical basis of our study is based on the pioneering work of Alsharari et al. [11], where the existence, uniqueness, and stability of UHS solutions of $\vartheta$-FSIEs were analyzed with high precision. In this work, the authors relied on advanced techniques of stochastic calculus and Banach Fixed Point Theory (BFPT) to prove their theoretical results. The equation studied takes the following form:

$$\begin{array}{cc}\hfill \alpha \left(\omega \right)=& \eta +{\displaystyle \underset{a}{\overset{\omega }{\int }}}{\mathrm{\Psi }}_1(\varsigma ,\alpha \left(\varsigma \right))d\varsigma +{\displaystyle \sum _{i=2}^n}{\displaystyle \underset{a}{\overset{\omega }{\int }}}{\vartheta }_i^{\prime }\left(\varsigma \right){\left({\vartheta }_i\left(\omega \right)-{\vartheta }_i\left(\varsigma \right)\right)}^{{\gamma }_i-1}{\mathrm{\Psi }}_i(\varsigma ,\alpha \left(\varsigma \right))d\varsigma \hfill \\ \hfill & +{\displaystyle \underset{a}{\overset{\omega }{\int }}}F(\varsigma ,\alpha \left(\varsigma \right))dW\left(\varsigma \right),\hfill \end{array}$$

(1)

with $\eta \in \mathbb{R}$, ${\gamma }_i\in \left(0,1\right)$ for $2\le i\le n$, $\omega \in [a,T]$ and specific conditions were imposed on the functions F and ${\mathrm{\Psi }}_i$ to ensure Lipschitz continuity and bounds, which ensures the stability and accuracy of the solutions [11]. Despite the theoretical power of this analysis, the practical applications of these equations, especially $\vartheta$-FSIEs, in modeling realistic phenomena such as tumor growth and chemical diffusion in biological tissues remain insufficiently explored. This study seeks to fill this gap by developing an integrated computational framework that aims to test and apply $\vartheta$-FSIEs in such biological contexts. This framework is based on combining stochastic and fractional properties with biological tissue dynamics, enabling more accurate simulation of complex phenomena involving cell–chemical interactions. The numerical simulation involves the use of advanced numerical techniques to solve $\vartheta$-FSIEs, with an emphasis on the stability and accuracy of the solutions to ensure that the behaviors of biological systems are reliably captured. Through this work, we aim to provide new insights into the applicability of these equations in areas such as the design of targeted therapies and the analysis of the diffusion of chemicals in injured tissues, contributing to the development of more effective mathematical models for biomedical research. Equation (1) was selected as a mathematical model for tumor growth and chemical diffusion due to its ability to represent the complex dynamics of these biological phenomena. This equation relies on three key components that reflect the multiscale nature of the system under study: (1) the kernel-dependent integral terms $\vartheta$, which incorporate the historical effects of interactions between tumor cells and their environment, allowing for the description of delayed chemical diffusion in biological tissues and memory-dependent diffusion; (2) the stochastic terms $dW\left(\varsigma \right)$, which represent environmental fluctuations such as random changes in nutrient supply and genetic mutations in cancer cells; (3) the ${\gamma }_i$, which provides an accurate description of anomalous diffusion patterns, describing slow diffusion (subdiffusion) in dense tumor environments and rapid diffusion in areas of high vascular density. This mathematical structure enables the simulation of transitions between different diffusion patterns documented in experimental studies and aids in analyzing tumor responses to environmental changes. The proposed model is based on mathematical principles designed to capture key aspects of tumor growth and diffusion processes. It provides a robust theoretical framework for studying tumor dynamics and serves as a foundation for future research. To strengthen the clinical relevance of this framework, our next steps involve collaborating with oncologists and experimental biologists to integrate tissue-specific data (e.g., MRI tumor measurements or in vitro diffusion rates). This collaboration will enable direct model calibration and validation, bridging the gap between theoretical predictions and real-world biomedical challenges, and ultimately enhancing the model’s predictive power and applicability to real-world tumor dynamics.

The main contribution of this paper is to present an advanced numerical simulation of tumor growth and chemical diffusion using $\vartheta$-FSIEs. Through this work, we highlight the dual role of fractional and stochastic dynamics in shaping complex biological patterns, providing a deeper understanding of the interaction of physical and biological factors in biological tissue environments. These simulations provide a detailed insight into the influence of fractional time and stochastic perturbations in controlling tumor growth dynamics and chemical diffusion. In particular, $\vartheta$-FSIEs allow an accurate representation of the transitions between subdiffusion and superdiffusion, reflecting the multi-scale nature of these biological processes. By combining a robust theoretical framework with accurate numerical simulations, this study opens up new avenues for employing $\vartheta$-FSIEs in modeling complex biological systems, further enhancing our contribution to the fields of biomedicine and mathematical computation.

The rest of this paper is organized as follows: Section 2 reviews the main theoretical results obtained. Section 3 presents the numerical results, focusing on the simulation of tumor growth and the diffusion of chemicals in biological tissues. This section demonstrates how $\vartheta$-FSIEs can be applied to realistic models, reviewing the numerical strategies used and their performance in terms of accuracy and stability. The results are analyzed in detail to illustrate the influence of randomness and fractional dynamics on biological processes. Section 4 concludes by providing basic insights into the significance of the study and the results achieved, and discussing the future potential of $\vartheta$-FSIEs in biological modeling. New research directions are proposed, including improving the numerical models and exploring other applications in biophysics and complex stochastic systems.

2. Existence, Uniqueness, and Stability Analysis of $\vartheta$-FSIEs

The seminal work by Alsharari et al. (2025, [11]) focuses on establishing accurate results for a class of $\vartheta$-FSIEs, representing an important advance in analyzing complex systems that combine stochastic effects and fractional dynamics. The primary motivation for this work stems from the need for more accurate and flexible mathematical models capable of describing systems that exhibit memory-dependent behaviors and nonlinear diffusive effects, properties that traditional methods often fail to capture adequately. By adopting a robust mathematical framework that combines fractional calculus and stochastic analysis, the authors address fundamental questions regarding the existence and uniqueness of solutions to $\vartheta$-FSIEs. These results are essential for understanding whether complex systems can be accurately modeled using this class of equations. Furthermore, the work focuses on the stability of solutions, including UHS, which ensures that systems respond to small perturbations in the initial conditions, making the models more robust and applicable in real-world contexts. These results provide a solid foundation for future research into applying $\vartheta$-FSIEs in new fields and more complex real-world problems. By providing this in-depth analysis, Alsharari et al. contribute to the theoretical understanding of $\vartheta$-FSIEs, opening up broad prospects for their application in a variety of complex dynamical systems.

The paper [11] provides important theoretical progress expressed through three main theories. Together, these theories represent an integrated framework for analyzing and studying $\vartheta$-FSIEs, opening new horizons for their application in various fields, which we summarize and analyze below.

Lemma 1 

(Operator definition). The following assumptions are made:

A1. 

Lipschitz condition:

The functions F, ${\mathrm{\Psi }}_l$ for $1\le l\le n$ satisfy the Lipschitz condition:

$$\left|F(\omega ,u_1)-F(\omega ,u_2)\right|\vee \left|{\mathrm{\Psi }}_l(\omega ,u_1)-{\mathrm{\Psi }}_l(\omega ,u_2)\right|\le K\left|u_1-u_2\right|,$$

for all $\omega \in [a,T]$, $u_1,u_2\in {\mathbb{R}}^2$, with $K>0$.

A2. 

Boundedness condition:

The functions F, ${\mathrm{\Psi }}_l$ for $1\le l\le n$ satisfy:

$$\underset{\omega \in [a,T]}{esssup}\left|F(\omega ,0)\right|\vee \underset{\omega \in [a,T]}{esssup}\left|{\mathrm{\Psi }}_l(\omega ,0)\right|\le d,$$

for some $d>0$, where $1\le l\le n.$

Then the operator $N_{\eta }$ is well-defined for every $\eta \in \mathbb{R}$, where the operator $N_{\eta }$ is defined on the space of stochastic processes $S_m\left([a,T]\right)$, which represents the set of all measurable and $\mathcal{M}$-adapted processes satisfying

$$\underset{\omega \in [a,T]}{sup}{\parallel \alpha \left(\omega \right)\parallel }_m<\infty .$$

More precisely, this space is a Banach space equipped with the norm:

$${\parallel \alpha \parallel }_{S_m\left([a,T]\right)}=\underset{\omega \in [a,T]}{sup}{\parallel \alpha \left(\omega \right)\parallel }_m.$$

The operator $N_{\eta }$ is explicitly defined as

$$\begin{array}{cc}\hfill \left(N_{\eta }\alpha \right)\left(\omega \right)=& \eta +{\int }_a^{\omega }{\mathrm{\Psi }}_1(\varsigma ,\alpha \left(\varsigma \right))d\varsigma \hfill \\ \hfill & +\sum _{i=2}^n{\int }_a^{\omega }{\vartheta }_i^{\prime }\left(\varsigma \right){({\vartheta }_i\left(\omega \right)-{\vartheta }_i\left(\varsigma \right))}^{{\gamma }_i-1}{\mathrm{\Psi }}_i(\varsigma ,\alpha \left(\varsigma \right))d\varsigma \hfill \\ \hfill & +{\int }_a^{\omega }F(\varsigma ,\alpha \left(\varsigma \right))dW\left(\varsigma \right).\hfill \end{array}$$

This preliminary lemma ensures the mathematical soundness of the operator central to the analysis.

The first theorem ensures that the model is mathematically consistent and able to predict accurately, a property essential for applications in engineering and science. This consistency is an indication that the equations used in the model have unique, existing solutions under the assumed conditions, removing any mathematical ambiguity that might lead to unreliable results. On a practical level, this assurance enhances the reliability of the model when used to simulate complex physical and biological systems. For example, in the biological sciences, proving the existence and uniqueness of solutions is essential for modeling biological processes such as tumor growth and the diffusion of chemicals in tissues, where these applications require accurate decisions based on robust mathematical models. Furthermore, this predictive consistency enhances the model’s ability to explore unfamiliar scenarios, enabling it to be used as a tool for exploring nonlinear phenomena in fields such as biophysics and environmental modeling. This theoretical foundation paves the way for improving the model and extending it to other, more complex applications, enhancing its value in scientific and engineering research.

Theorem 1 

(Existence and uniqueness). Under the assumptions A1 and A2, the $\vartheta$-FSIE (1) is guaranteed to have a unique solution.

The second theorem proves that $\vartheta$-FSIE (1) solutions exhibit a continuous dependence on initial conditions, meaning that small changes in the initial conditions lead to small and proportional changes in the solutions. This property is pivotal in mathematical modeling, as it reflects the stability and predictability of the system. This property enhances the stability and robustness of the solutions under small perturbations in the initial conditions. In biological applications, such as modeling chemical diffusion or tumor dynamics, this property means that model-based predictions are reliable even when there is a small uncertainty in the initial values, such as the initial chemical concentration or the distribution of cancer cells. Furthermore, this property is a fundamental step towards UHS analysis, which provides deeper insights into the long-term dynamic behavior of the system, making the model more suitable for practical applications that require accurate and consistent responses under small environmental or cognitive changes.

Theorem 2 

(Continuous dependence on initial conditions [11]). For two initial values ${\eta }_1$, ${\eta }_2\in \mathbb{R}$, the difference between their respective solutions ${\alpha }_1\left(\varsigma \right)$ and ${\alpha }_2\left(\varsigma \right)$ of (1) vanishes as ${\eta }_1⟶{\eta }_2$:

$$\underset{{\eta }_1⟶{\eta }_2}{lim}\underset{\varsigma \in [a,T]}{sup}{∥{\alpha }_1\left(\varsigma \right)-{\alpha }_2\left(\varsigma \right)∥}_{ms}=0.$$

The latter theorem proves that the $\vartheta$-FSIE (1) is stable according to the UHS criterion. This property indicates that the approximate solutions, resulting from small disturbances or errors in the initial data, remain close to the exact solutions. This stability is an important indicator of the robustness and reliability of the model, especially in practical applications where errors and approximations are inevitable. This property plays a crucial role in ensuring that the model can tolerate small deviations without significantly affecting the accuracy of its predictions. For example, in numerical simulations of biological or physical processes, such as tumor growth or chemical diffusion, UHS stability ensures that errors due to numerical slicing, computational approximations, or experimental measurements do not lead to significant deviations in the results. Furthermore, UHS stability provides a solid basis for the design of efficient numerical algorithms, as it ensures that the approximations resulting from numerical solutions will not deviate significantly from the theoretical solutions. This is of great importance in complex systems where fractional dynamics interact with stochastic effects, posing a challenge for accurate modeling and reliable simulation. The applications of UHS also extend to dynamical systems, optimal control problems, and model-based decision-making, where this stability lends confidence in the reliability of predictions and their relevance to real behaviors. Thus, the UHS stability of the $\vartheta$-FSIE (1) equation enhances the mathematical framework of the model and improves its practical applicability in various scientific and engineering fields.

Theorem 3 

(Ulam–Hyers stability). If A1 and A2 are satisfied, then the $\vartheta$-FSIE (1) is UHS.

The results presented by Alsharari et al. [11] represent a major step forward in the theory of fractional stochastic systems. By taking advantage of continuity conditions and Lipschitz boundary, the authors provide a powerful mathematical framework for studying $\vartheta$-FSIEs. The theorems presented ensure that these equations not only have unique and well-behaved solutions but also have desirable stability properties, such as UHS. These results are pivotal for the practical application of fractional stochastic models, as they provide accurate and reliable tools for analyzing complex systems that combine fractional and stochastic effects. This is particularly evident in areas such as control theory, where control of fine-grained dynamical systems is required, signal processing, where non-local dynamics play a major role in signal optimization, and biological systems, where these models enable a deeper understanding of biological interactions involving diffusion and stochastic processes. Furthermore, these results provide insights into how to balance accurate modeling and numerical considerations in practical applications. The ability to prove the existence and uniqueness of solutions, along with stability properties, makes it possible to build models that can deal with the actual challenges associated with volatility and uncertainty in systems.

3. Numerical Simulation and Analysis of Tumor Growth Dynamics

Understanding tumor growth and the dynamics of chemical diffusion within biological tissues requires advanced simulation techniques and powerful analytical frameworks. In this context, simulations are built based on Equation (1). Through this complex model, how the tumor evolves and how environmental and therapeutic factors affect this growth are determined. This model uses numerical integration techniques to simulate the complex interactions between tumor cells and their surrounding chemicals, and the effects of chemotherapies and radiation. The process begins by simulating tumor evolution using this equation, and subsequently progresses to evaluate the different effects of treatments by simulating chemical interactions with therapeutics and using neural networks to recognize patterns and predict future tumor evolution. We also use predictive modeling techniques to enhance our understanding of how the tumor responds to the environment and treatment, which contributes to providing deeper insights into its behavior under multiple conditions. These numerical simulations provide powerful tools to support therapeutic decision-making and guide future research in oncology.

Figure 1 illustrates the natural growth trajectory of a tumor in the absence of any treatment, revealing a typical exponential growth pattern characterized by a steady increase in tumor size over time. This exponential growth reflects the process of accelerated tumor cell proliferation, where the tumor continues to expand at an increasing rate when there are no therapeutic interventions to stop it. This pattern is a critical parameter in studying the behavior of untreated tumors, highlighting the aggressive nature of the tumor and its ability to spread rapidly and widely into neighboring tissues. These data do not emphasize the importance of early detection of tumors and initiating appropriate therapeutic interventions as soon as possible to prevent this uncontrolled increase in tumor size. Although this exponential growth indicates a relatively early stage of tumor development, tumor development in the absence of treatment remains an essential starting point for understanding tumor dynamics and how biological factors influence its response to treatment.

Figure 2 compares tumor growth under three different treatment conditions: no treatment, chemotherapy, and placebo. The “no treatment” curve replicates the baseline from Figure 1, reflecting the steady exponential growth pattern that characterizes an untreated tumor. This pattern shows a continuous increase in tumor size over time without intervention, reflecting the aggressive and rapid growth nature of malignant tumors in the absence of treatment. This condition serves as an important reference for evaluating the effectiveness of different treatments. In contrast, the “chemotherapy” curve shows a significant decrease in growth rate, indicating that chemotherapy has succeeded in significantly slowing tumor growth. Although chemotherapy reduces tumor growth, this decrease is not always sufficient to completely stop tumor progression, highlighting the limitations of some chemotherapies in controlling growth in the long term. Nevertheless, chemotherapy remains an effective treatment option in many cases requiring a rapid response. The “placebo” curve shows a more pronounced suppression of tumor growth than chemotherapy, suggesting that a placebo may provide superior benefits in managing tumor dynamics. Alternative treatment may include the use of innovative therapeutic techniques or therapeutic combinations specifically developed to attack the tumor in different ways than conventional treatments. This treatment shows a strong effect in reducing tumor size in the long term, which opens the door to research into its potential as a more effective alternative to some chemotherapies, especially in cases that do not respond to conventional drugs. This comparison contributes to shedding light on the relative effectiveness of different interventions in managing tumor dynamics, providing deeper insights into optimal treatment strategies. Through this analysis, the most effective treatments can be identified based on their different effects in reducing tumor growth, thus improving future treatment strategies. This comparison also provides a basis for developing personalized therapeutic frameworks based on tumor type and response to treatment, which could enhance the improvement of patient care.

Figure 3 illustrates the ability of the neural network model to recognize and replicate observed tumor growth patterns. By comparing the observed data with the model’s predictions, it is clear that the neural network accurately captures the underlying dynamics of tumor growth, including exponential patterns and changes in tumor size over time. The model has a high ability to predict tumor progression under different conditions, whether untreated or multi-treatment, enhancing the accuracy of predictive understanding of tumor behavior. The strength of neural networks is their ability to learn from large and complex data sets, making them an ideal tool for analyzing biological patterns that may be hidden or complex to traditional methods. The network’s ability to adapt to observed data and generate accurate results makes it a reliable tool for understanding tumor growth processes and their interactions with different treatments. Validation of the model by comparing predictions with actual data shows that the neural network captures not only general trends in tumor growth but also subtle responses to changes in therapeutic parameters. This reliability strengthens the position of neural networks as powerful analytical tools for understanding complex, multidimensional biological processes. Not only does this analysis provide deeper insights into tumor behavior under different influences, it also paves the way for the use of neural networks in predictive modeling, allowing researchers and clinicians to predict future tumor evolution based on current data. These networks can then be used to guide advanced treatment strategies, providing accurate predictions that support better, scientifically informed treatment decisions.

Figure 4 illustrates the ability of the neural network to predict future tumor growth, expanding the focus to predict long-term tumor evolution dynamics. The neural network allows growth trends to be extrapolated from observed data and accurately predicted over extended periods. The predicted values generated by the network closely match the actual data, highlighting the power of the model in simulating and identifying potential tumor growth trajectories under a variety of conditions. The results show that the neural network is not limited to processing current data, but can also provide accurate estimates of the future based on past patterns, providing an invaluable tool in estimating future tumor evolution. These predictive capabilities are invaluable in clinical decision-making, as clinicians and researchers can use them to proactively identify potential tumor growth trends. With this predictive ability, personalized treatment plans that take into account potential tumor evolution can be designed, contributing to more personalized and effective treatments for each patient. In addition, this ability allows changes in tumor status to be anticipated, enabling early therapeutic interventions to prevent progression or improve treatment efficacy at different stages of growth. Overall, the importance of neural networks lies in providing data-driven prediction solutions, which contribute to dynamically guiding treatment based on a deep understanding of the biological behavior of tumors. The ability of these networks to predict tumor growth opens a new horizon in data-driven medical treatment, which enhances the effectiveness of clinical procedures and directs them toward the most positive outcomes.

Remark 1. 

In this paper, we performed numerical simulations using the following schemes: (1) a fixed-time-step Euler forward method ($\Delta t=1$) for the logistic tumor growth model, (2) adaptive numerical integration using the integral() function in MATLAB (https://www.mathworks.com/products/matlab.html) to model chemical diffusion, and (3) a prediction model using support vector machines (SVMs) for long-term prediction. To ensure the accuracy of the results, we performed a sensitivity analysis by varying the growth parameters (0.03–0.07) and diffusion parameters (0.6–1.0), and verified the stability of the solutions as the grid resolution changed (n = 100 to n = 1000). The results demonstrated numerical stability with a less than 5% relative change in the main variables, while the prediction model achieved an accuracy with a mean squared error of 0.05. This methodology strikes a balance between computational accuracy and computational efficiency over the studied parameter range.

Remark 2. 

The ϑ-FSIEs-based model offers clear competitive advantages over traditional models used to simulate tumor growth and chemical diffusion in biological tissues. First, the incorporation of stochasticity and fractional effects allows for a more accurate representation of nonlinear biological phenomena, which is critical for the study of complex biological systems. Second, simulations have shown that the model offers high numerical stability, with the relative error reduced to less than 5%when varying growth and chemical diffusion parameters, enhancing the reliability of the results. Furthermore, the model is flexible in incorporating different biological parameters and can be easily adjusted to simulate multiple scenarios, including the effects of chemotherapy and tumor–environment interactions. Finally, this approach offers improved computational efficiency. This modeling framework not only contributes to a better theoretical understanding of tumor progression mechanisms but can also serve as a practical tool for clinical decision-making by providing more accurate predictions of tumor response to various treatments.

4. Conclusions

This paper presents an advanced framework for simulating tumor growth and chemical diffusion in biological tissues using $\vartheta$-FSIEs. By combining theoretical analysis and numerical simulation, we explored the dynamic interactions between fractional and stochastic effects in complex biological environments. The results highlight the importance of fractional time and stochastic perturbations in shaping tumor growth dynamics and chemical transport processes, providing new insights into the applicability of $\vartheta$-FSIEs in modeling biological phenomena such as tumor progression and therapeutic interventions. Incorporating fractional effects into biological modeling provides an additional layer of accuracy in simulating biological systems characterized by random and nonlinear behavior, a common feature in tumor progression and chemical diffusion. This model allows the study of the effects of small perturbations on the system, helping to understand how small changes in parameters affect complex biological dynamics. Simulations demonstrated the ability of the $\vartheta$-FSIE model to replicate realistic tumor growth patterns and model the diffusion of chemicals in biological tissues with high accuracy. This framework also provided valuable insights into the effects of different treatment strategies, especially chemotherapy. Through comparative analyses, we observed the potential of alternative therapies to suppress tumor growth over long periods, opening new avenues for research into therapeutic strategies. Furthermore, by incorporating neural network-based predictive modeling, we enhanced our understanding of tumor growth under different conditions, demonstrating the power of these networks to accurately predict tumor progression. The ability of neural networks to recognize complex patterns and predict future growth trajectories adds a crucial layer to the modeling framework, providing valuable support in decision-making about treatment strategies. These networks may also provide support in evaluating the efficacy of future therapies by simulating various treatment scenarios. Additionally, simulations of tumor growth under various conditions, including untreated scenarios and treatment-based interventions, provide an application case that validates the model’s predictive capabilities. The primary tumor growth (Figure 1) is consistent with the observed exponential growth patterns, supporting the model’s accuracy in simulating real-world tumor dynamics. Furthermore, a comparative analysis of chemotherapy and alternative therapies (Figure 2) highlights the model’s ability to assess treatment efficacy, making it a valuable tool for evaluating therapeutic strategies. Predictive modeling using neural networks (Figure 3 and Figure 4) enhances the model’s potential for clinical decision-making, demonstrating its ability to predict tumor progression under various treatment regimens.

In conclusion, the combination of stochastic and fractional dynamics enables a more comprehensive understanding of tumor behavior and chemokinesis, contributing to the development of more effective mathematical models for biomedical applications. The use of this model could enhance the accuracy of predictions about tumor progression and guide research efforts toward personalized therapeutic approaches. Next steps will prioritize improving numerical methods and exploring their potential in other areas of biophysics and complex systems, particularly in targeted therapy design and chemotherapy-driven tumor dynamics. The model’s flexibility also allows extensions to scenarios like tumor–vascular interactions or environmental–genetic effects.

To strengthen clinical relevance, we plan to collaborate with experimental biologists for tissue-specific data (e.g., MRI tumor measurements or in vitro diffusion rates), enabling direct model calibration and validation. This will bridge the gap between theoretical predictions and real-world biomedical challenges.