Mathematical Model Of Tumor Growth Research Articles

Mathematical Model of Cancer with Ordinary Differential Equations

The cancer cell starts dividing, infiltrating neighboring tissues and traveling throughout the lymphatic system. While there are ways to stop the spread of disease or get rid of infected cells, most of the approaches are unable to identify the early warning indicators of such an occurrence. Using diverse types of differential equations, especially ordinary differential equations (ODEs), is a helpful catalyst that experts employ. Using differential equations, researching resistance to chemotherapy, forecasting potential treatment failure, or evaluating the result and prognosis following various forms of therapy. Living things always include cancer cells, but a biological regulatory system keeps them from spreading to a dangerous degree (think overpopulation vs. natural resources). Therefore, the most efficient method of determining when to effectively intervene with the tumor growth is to use the cytokinetic method of quantitatively assessing the cancer cells’ progression. Cancer Metabolism: One of the main characteristics of cancer is metabolic reprogramming, in which the metabolism of cancer cells is changed to fuel their explosive growth and multiplication. New models of cancer metabolism investigate the roles that dysregulated metabolic pathways play in the genesis and spread of tumors, providing prospective avenues for therapeutic intervention. The mathematical models of tumor growth modeling of ordinary differential equations (ODEs cancer). The tumor grows voraciously, and the scientists and mathematicians who tried to have a better understanding grow. The study of such treatments on models of tumor growth leads to one or more ODEs. Which gives some ideas on the relation between equations and tumor growth in cancer cells introduce ODEs to provide mathematical models of tumor growth. The dynamics of tumor cells and their growths through clinical, experimental, and theoretical approaches, new ideas for different cancer therapies are developed with the goal of controlling and reducing the death rate for earlier diagnosis. The kinetics of tumor cell proliferation and its treatment approach were covered in this research. In order to comprehend the proliferation of tumor cells, we expanded the study and xamined a feew basic mathematical models.

Deep learning for autosegmentation for radiotherapy treatment planning: State-of-the-art and novel perspectives.

The rapid development of artificial intelligence (AI) has gained importance, with many tools already entering our daily lives. The medical field of radiation oncology is also subject to this development, with AI entering all steps of the patient journey. In this review article, we summarize contemporary AI techniques and explore the clinical applications of AI-based automated segmentation models in radiotherapy planning, focusing on delineation of organs at risk (OARs), the gross tumor volume (GTV), and the clinical target volume (CTV). Emphasizing the need for precise and individualized plans, we review various commercial and freeware segmentation tools and also state-of-the-art approaches. Through our own findings and based on the literature, we demonstrate improved efficiency and consistency as well as time savings in different clinical scenarios. Despite challenges in clinical implementation such as domain shifts, the potential benefits for personalized treatment planning are substantial. The integration of mathematical tumor growth models and AI-based tumor detection further enhances the possibilities for refining target volumes. As advancements continue, the prospect of one-stop-shop segmentation and radiotherapy planning represents an exciting frontier in radiotherapy, potentially enabling fast treatment with enhanced precision and individualization.

Mathematical Modeling of Tumor Growth in Preclinical Mouse Models with Applications in Biomarker Discovery and Drug Mechanism Studies.

Oncology drug efficacy is evaluated in mouse models by continuously monitoring tumor volumes, which can be mathematically described by growth kinetic models. While past studies have investigated various growth models, their reliance on small datasets raises concerns about whether their findings are truly representative of tumor growth in diverse mouse models under different vehicle or drug treatments. Here, we systematically evaluated six parametric models (exponential, exponential quadratic, monomolecular, logistic, Gompertz, and von Bertalanffy) and the semi-parametric generalized additive model (GAM) on fitting tumor volume data from over 30,000 mice in 930 experiments conducted in patient-derived xenografts, cell line-derived xenografts, and syngeneic models. We found that the exponential quadratic model is the best parametric model and can adequately model 87% studies, higher than other models including von Bertalanffy (82%) and Gompertz (80%) models, the latter is often considered the standard growth model. On the mouse group level, 7.5% of growth data could not be fit by any parametric model and were fitted by GAM. We show that eGaIT, a GAM derived efficacy metric, is equivalent to eGR, a metric we previously proposed and conveniently calculated by simple algebra. Using five studies on Paclitaxel, anti-PD-1 antibody, Cetuximab, Irinotecan, and Sorafenib, we show that exponential and exponential quadratic models achieve similar performance in uncovering drug mechanism and biomarkers. We also compared eGR-based association analysis and exponential modeling approach in biomarker discovery and found they complement each other. Modeling methods herein are implemented in an open-source R package freely available at https://github.com/hjzhou988/TuGroMix.

Simulation of Malignant Tumor Growth through a Macroscopic Approach Model

A tumor refers to an abnormal cell cluster resulting from uncontrolled cell division, manifesting as growths within bodily organs and exhibiting rapid proliferation. Tumors are categorized into benign and malignant types, influenced by various factors such as viral infections, chemical exposure, and genetic predisposition. Mathematical modeling of tumor growth often involves differential equations. This study represents tumors as spatial regions in R2 with free boundaries, utilizing the Hele-Shaw model from fluid dynamics. Malignant tumor cell progression is simulated mathematically, employing a macroscopic model to ascertain cell characteristics and growth stages. Additionally, the discussion highlights the limitation of the traveling wave solution in describing tumor development solely as an incompressible fluid, leading to the adoption of the Hele-Shaw model in this research.

Mathematical modeling of tumor growth as a random process in the presence of interaction between tumor cells and normal cells

A mathematical model of tumor growth as a random process is studied, in which there is an interaction between tumor cells and normal cells. A set of two coupled stochastic differential equations defines the dynamics of tumor cells and normal cells. The evolution contains driven growth, therapy, and Gaussian white noises terms. The corresponding Fokker–Planck equation is used to study the large-time behavior of the system. Specifically, for periodic therapy terms, the ratio of the probability of large tumor cells number to the probability of small tumor cells number is investigated. It is seen that increasing the interaction between the tumor cells and normal cells decreases this ratio.

A computational tumor growth model experience based on molecular dynamics point of view using deep cellular automata

Cancer, as identified by the World Health Organization, stands as the second leading cause of death globally. Its intricate nature makes it challenging to study solely based on biological knowledge, often leading to expensive research endeavors. While tremendous strides have been made in understanding cancer, gaps remain, especially in predicting tumor behavior across various stages. The integration of artificial intelligence in oncology research has accelerated our insights into tumor behavior, right from its genesis to metastasis. Nevertheless, there's a pressing need for a holistic understanding of the interactions between cancer cells, their microenvironment, and their subsequent interplay with the broader body environment. In this landscape, deep learning emerges as a potent tool with its multifaceted applications in diverse scientific challenges. Motivated by this, our study presents a novel approach to modeling cancer tumor growth from a molecular dynamics' perspective, harnessing the capabilities of deep-learning cellular automata. This not only facilitates a microscopic examination of tumor behavior and growth but also delves deeper into its overarching behavioral patterns. Our work primarily focused on evaluating the developed tumor growth model through the proposed network, followed by a rigorous compatibility check with traditional mathematical tumor growth models using R and Matlab software. The outcomes notably aligned with the Gompertz growth model, accentuating the robustness of our approach. Our validated model stands out by offering adaptability to diverse tumor growth datasets, positioning itself as a valuable tool for predictions and further research.

Bifurcation and Stability of a Mathematical Model for Tumor Growth with Oncolytic Virotherapy

Bifurcation and Stability of a Mathematical Model for Tumor Growth with Oncolytic Virotherapy

Optimization of Size of Nanosensitizers for Antitumor Radiotherapy Using Mathematical Modeling.

The efficacy of antitumor radiotherapy can be enhanced by utilizing nonradioactive nanoparticles that emit secondary radiation when activated by a primary beam. They consist of small volumes of a radiosensitizing substance embedded within a polymer layer, which is coated with tumor-specific antibodies. The efficiency of nanosensitizers relies on their successful delivery to the tumor, which depends on their size. Increasing their size leads to a higher concentration of active substance; however, it hinders the penetration of nanosensitizers through tumor capillaries, slows down their movement through the tissue, and accelerates their clearance. In this study, we present a mathematical model of tumor growth and radiotherapy with the use of intravenously administered tumor-specific nanosensitizers. Our findings indicate that their optimal size for achieving maximum tumor radiosensitization following a single injection of their fixed total volume depends on the permeability of the tumor capillaries. Considering physiologically plausible spectra of capillary pore radii, with a nanoparticle polymer layer width of 7 nm, the optimal radius of nanoparticles falls within the range of 13-17 nm. The upper value is attained when considering an extreme spectrum of capillary pores.

Fractional calculus in mathematical oncology

Even though, nowadays, cancer is one of the leading causes of death, too little is known about the behavior of this disease due to its unpredictability from one patient to another. Classical mathematical models of tumor growth have shaped our understanding of cancer and have broad practical implications for treatment scheduling and dosage. However, improvements are still necessary on these models. The primary objective of the present research is to prove the efficiency of fractional order calculus in mathematical oncology, more specifically in tumor growth modeling. For this, a generalization of the four most used differential equation models in tumor volume measurements fitting is realized, using the corresponding fractional order equivalent. Are established the fractional order Exponential, Logistic, Gompertz, General Bertalanffy-Pütter and Classical Bertalanffy-Pütter models for a treated and untreated dataset. The obtained results are compared by Mean Squared Error (MSE) with the integer order correspondent of each model. The results prove the superiority of the fractional order models. The MSE of fractional order models are reduced at least at half in comparison with the MSE of the integer order equivalent. It is demonstrated in this way that fractional order deterministic models can offer a good starting point in finding a proper mathematical model for tumor evolution prediction. Fractional calculus is a suitable method in this case due to its memory property, aspect that particularly characterizes biological processes.

Multiscale modeling of the dynamic growth of cancerous tumors under the influence of chemotherapy drugs

A comprehensive model of chemotherapy treatment of cancer can help us to optimize the drug administration/dosage and improve the treatment outcome. In the present study, a multiscale mathematical model of tumor growth during chemotherapy treatment is developed to predict its response to the medication and cancer progression. The modeling is a continuous multiscale simulation consisting of three tissue phases including cancer cells, normal cells, and extracellular matrix. In addition to the drug administration, the impacts of immune cells, programmed cell death, nutrient competition, and glucose concentration are included. The outputs of our mathematical model conform to the published experimental and clinical data, and it can be used in optimizing chemotherapy, and personalized cancer treatment.

A dynamical model of the immune system interaction in a melanoma

Melanoma is a concerning variety of skin cancer which prevalence arises in our society. Its lethality relies on its ability to migrate and metastasize. In this paper, we use a mathematical model of tumor growth and lysis, and we adjust it to model melanoma growth. Tumor growth is modeled through the action of a cellular automaton and two reaction–diffusion equations for nutrients distribution. In order to represent the melanoma, we have induced mutations in random cells, specifically in their reproductive capacity, so that tumor growth appears less structured and consistent. A coherent mathematical model for the decay of the melanoma has been created by means of an exponential function. The results show that the inner morphology plays an important role in cancer lysis: as the complexity increases, so does lysis time. We have also compared simulations to real melanomas with the aim of acknowledging the robustness of the melanoma modeling.

Interaction of Virus in Cancer Patients: A Theoretical Dynamic Model.

This study reports on a phase-space analysis of a mathematical model of tumor growth with the interaction between virus and immune response. In this study, a mathematical determination was attempted to demonstrate the relationship between uninfected cells, infected cells, effector immune cells, and free viruses using a dynamic model. We revealed the stability analysis of the system and the Lyapunov stability of the equilibrium points. Moreover, all endemic equilibrium point models are derived. We investigated the stability behavior and the range of attraction sets of the nonlinear systems concerning our model. Furthermore, a global stability analysis is proved either in the construction of a Lyapunov function showing the validity of the concerned disease-free equilibria or in endemic equilibria discussed by the model. Finally, a simulated solution is achieved and the relationship between cancer cells and other cells is drawn.

Stability analysis of a targeted chemotherapy-cancer model

<abstract><p>This study addresses a modified mathematical model of tumor growth with targeted chemotherapy consisting of effector cells, tumor cells, and normal cells. To investigate the dynamics of the model, local and global stability analyses have been performed at the equilibrium points of the model. It is found that the tumor-free steady state is globally asymptotically stable under certain conditions, which suggests that the prescribed treatment can eradicate tumor cells from the body for a threshold value of tumor growth rate. The main result of this study is that if the tumor growth rate is tiny, it is possible to eradicate the tumor from the body using a smaller amount of targeted chemotherapy drugs with less harm to the other healthy cells. If not, it requires a high dose of targeted chemotherapy drugs, which can increase the side effects of the drugs. Numerical simulations have been performed to verify our analytical results.</p></abstract>

Mathematical modeling of tumor growth and treatment: Triple negative breast cancer

In this study, a mathematical model of triple negative breast cancer (TNBC) with checkpoint inhibitor immunotherapy is proposed. The mathematical model includes interactions between tumor cells, innate and adaptive immune cells, programmed cell death protein 1 (PD-1), programmed death ligand 1 (PD-L1), and immune checkpoint inhibitor (ICI). The mathematical model exhibits two coexisting stable equilibria, representing the tumor-free and the large-tumor states. While the tumor-free equilibrium is locally unstable, mathematical analysis and numerical simulation prove that the model immune system is able to eliminate a tiny tumor, providing immune surveillance. Further simulation shows that natural kill (NK) cell response is able to effectively eliminate tumor cells and PD-L1 expression inhibits cytotoxic T lymphocyte (CTL) response. With immune checkpoint inhibitor, the simulated treatment is able to eliminate a larger tumor but the effectiveness is limited by the intrinsic immune response. Enhancing CTL response significantly improves the outcome of treatment. This suggests that a combination of checkpoint inhibitor and immune booster therapy can produce synergistic effect.

Dynamic characterization of breast cancer response to neoadjuvant therapy using biophysical metrics of spatial proliferation

Current tools to assess breast cancer response to neoadjuvant chemotherapy cannot reliably predict disease eradication, which if possible, could allow early cessation of therapy. In this work, we assessed the ability of an image data-driven mathematical modeling approach for dynamic characterization of breast cancer response to neoadjuvant therapy. We retrospectively analyzed patients enrolled in the I-SPY 2 TRIAL at the Atrium Health Wake Forest Baptist Comprehensive Cancer Center. Patients enrolled on the study received four MR imaging examinations during neoadjuvant therapy with acquisitions at baseline (T0), 3-weeks/early-treatment (T1), 12-weeks/mid-treatment (T2), and completion of therapy prior to surgery (T3). We use a biophysical mathematical model of tumor growth to generate spatial estimates of tumor proliferation to characterize the dynamics of treatment response. Using histogram summary metrics to quantify estimated tumor proliferation maps, we found strong correlation of mathematical model-estimated tumor proliferation with residual cancer burden, with Pearson correlation coefficients ranging from 0.88 and 0.97 between T0 and T2, representing a significant improvement from conventional assessment methods of change in mean apparent diffusion coefficient and functional tumor volume. This data shows the significant promise of imaging-based biophysical mathematical modeling methods for dynamic characterization of patient-specific response to neoadjuvant therapy with correlation to residual disease outcomes.

Mathematical Modeling and Analysis of Tumor Chemotherapy

Cancer diseases lead to the second-highest death rate all over the world. For treating tumors, one of the most common schemes is chemotherapy, which can decrease the tumor size and control the progression of cancer diseases. To better understand the mechanisms of chemotherapy, we developed a mathematical model of tumor growth under chemotherapy. This model includes both immune system response and drug therapy. We characterize the symmetrical properties and dynamics of this differential equation model by finding the equilibrium points and exploring the stability and symmetry properties in a range of model parameters. Sensitivity analyses suggest that the chemotherapy drug-induced tumor mortality rate and the drug decay rate contribute significantly to the determination of treatment outcomes. Numerical simulations highlight the importance of CTL activation in tumor chemotherapy.

Classical mathematical models for prediction of response to chemotherapy and immunotherapy.

Classical mathematical models of tumor growth have shaped our understanding of cancer and have broad practical implications for treatment scheduling and dosage. However, even the simplest textbook models have been barely validated in real world-data of human patients. In this study, we fitted a range of differential equation models to tumor volume measurements of patients undergoing chemotherapy or cancer immunotherapy for solid tumors. We used a large dataset of 1472 patients with three or more measurements per target lesion, of which 652 patients had six or more data points. We show that the early treatment response shows only moderate correlation with the final treatment response, demonstrating the need for nuanced models. We then perform a head-to-head comparison of six classical models which are widely used in the field: the Exponential, Logistic, Classic Bertalanffy, General Bertalanffy, Classic Gompertz and General Gompertz model. Several models provide a good fit to tumor volume measurements, with the Gompertz model providing the best balance between goodness of fit and number of parameters. Similarly, when fitting to early treatment data, the general Bertalanffy and Gompertz models yield the lowest mean absolute error to forecasted data, indicating that these models could potentially be effective at predicting treatment outcome. In summary, we provide a quantitative benchmark for classical textbook models and state-of-the art models of human tumor growth. We publicly release an anonymized version of our original data, providing the first benchmark set of human tumor growth data for evaluation of mathematical models.

Existence and Uniqueness of Almost Periodic Solution for a Mathematical Model of Tumor Growth

This article is concerned with a mathematical model of tumor growth governed by 2nd order diffusion equation . The source of mitotic inhibitor is almost periodic and time-dependent within the tissue. The system is set up with the initial condition C(r, 0) = C0(r) and Robin type inhomogeneous boundary condition . Under certain conditions we show that there exists a unique solution for this model which is almost periodic.

Bifurcation of a Mathematical Model for Tumor Growth with Angiogenesis and Gibbs–Thomson Relation

In this paper, a mathematical model for solid vascular tumor growth with Gibbs–Thomson relation is studied. On the free boundary, we consider Gibbs–Thomson relation which means energy is expended to maintain the tumor structure. Supposing that the nutrient is the source of the energy, the nutrient denoted by [Formula: see text] satisfies [Formula: see text] where [Formula: see text] is a constant representing the ability of the tumor to absorb the nutrient through its blood vessels; [Formula: see text] is concentration of the nutrient outside the tumor; [Formula: see text] is the mean curvature; [Formula: see text] denotes adhesiveness between cells and [Formula: see text] denotes the exterior normal derivative on [Formula: see text] The existence, uniqueness and nonexistence of radially symmetric solutions are discussed. By using the bifurcation method, we discuss the existence of nonradially symmetric solutions. The results show that infinitely many nonradially symmetric solutions bifurcate from the radially symmetric solutions.

Improving cancer treatments via dynamical biophysical models.

Despite significant advances in oncological research, cancer nowadays remains one of the main causes of mortality and morbidity worldwide. New treatment techniques, as a rule, have limited efficacy, target only a narrow range of oncological diseases, and have limited availability to the general public due their high cost. An important goal in oncology is thus the modification of the types of antitumor therapy and their combinations, that are already introduced into clinical practice, with the goal of increasing the overall treatment efficacy. One option to achieve this goal is optimization of the schedules of drugs administration or performing other medical actions. Several factors complicate such tasks: the adverse effects of treatments on healthy cell populations, which must be kept tolerable; the emergence of drug resistance due to the intrinsic plasticity of heterogeneous cancer cell populations; the interplay between different types of therapies administered simultaneously. Mathematical modeling, in which a tumor and its microenvironment are considered as a single complex system, can address this complexity and can indicate potentially effective protocols, that would require experimental verification. In this review, we consider classical methods, current trends and future prospects in the field of mathematical modeling of tumor growth and treatment. In particular, methods of treatment optimization are discussed with several examples of specific problems related to different types of treatment.