Mathematical Model For Growth Research Articles

Mathematical modeling of Taraba State population growth using exponential and logistic models

The study models and simulates the future population growth of Taraba State using two models: exponential and logistic. Population data from 1991 to 2019 was obtained from the National Population Commission, and the models’ parameters were estimated and implemented using Python. The results showed that the exponential model performed better than the logistic model, with a higher R-squared value. Both models predict a growing population for Taraba State, reaching 9.8 million people by 2055, emphasizing the need for proper planning and policies to accommodate this growth. The study recommends that the state and federal governments use these findings to develop comprehensive plans for managing the increasing population.

Economic analysis of global microalgae biomass energy potential

With the increasing demand for renewable energy, microalgae, as a renewable biomass energy, can fix carbon dioxide and have broad application prospects in alleviating the energy crisis and improving the environment. In this paper, the potential biomass of global microalgae is calculated based on the mathematical growth model of microalgae proposed by predecessors. Based on this, this study further uses Newton's gravity model as the basic model of economic analysis and calculates the economic potential coefficient of microalgae production in various regions of the world by using the data of the world's top 20 cities in terms of urban population and urban GDP in 2020. The study has obtained the current global unused land with the high economic value of large-scale microalgae production areas, such as western North America, northern Africa, and northwest China, etc., which can provide guidance for the future site selection and development of microalgae biomass energy.

ОПРЕДЕЛЕНИЕ ТОЧЕК БИФУРКАЦИИ В ФУНКЦИОНИРОВАНИИ СИСТЕМЫ «ПОЧВА – РАСТЕНИЕ – ВОЗДУХ»

To date, a number of mathematical models of plant growth, developed by domestic and foreign scientists, are known. However, the issues of determining the bifurcation points that arise during the functioning of “soil-plant-air” system have not been sufficiently considered. In relation to the issues considered in the article, the bifurcation point is a critical state of the “plant” subsystem, at which it becomes unstable with respect to fluctuations in natural and climatic conditions (drought, frost, prolonged rains, etc.) and there is uncertainty in the development of plants (further growth or their death), as well as the intensive growth of plants as a result of the corresponding technological operations. For control and operational management of the formation of agricultural crops, it is desirable to know the bifurcation points determined by the biological time of plant growth and extreme weather situations. Therefore, the main goal of research is the analytical determination of bifurcation points observed during the vegetation of plants. The “plant” subsystem at the bifurcation point can be simultaneously in two or more states. As a result of the analysis of the obtained analytical dependences of various possible states of “soil-plant-air” system, it is proposed to subdivide bifurcations into negative and positive ones. A method has been obtained for determining bifurcation points during the functioning of “soil-plant-air” system. Under natural and climatic conditions, critical situations can arise with insufficient incoming substances to “soil-plant-air” system, such as light supply, moisture supply, heat supply, food supply and gas supply of plants. As a result of analytical studies, bifurcation points were determined in “soil-plant-air” system, depending on the radiation balance (R) formed on the underlying surface, slopes of different exposure and steepness, and coefficients characterizing light, moisture, heat, food and gas supply plants.

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.

Data driven model discovery and interpretation for CAR T-cell killing using sparse identification and latent variables.

In the development of cell-based cancer therapies, quantitative mathematical models of cellular interactions are instrumental in understanding treatment efficacy. Efforts to validate and interpret mathematical models of cancer cell growth and death hinge first on proposing a precise mathematical model, then analyzing experimental data in the context of the chosen model. In this work, we present the first application of the sparse identification of non-linear dynamics (SINDy) algorithm to a real biological system in order discover cell-cell interaction dynamics in in vitro experimental data, using chimeric antigen receptor (CAR) T-cells and patient-derived glioblastoma cells. By combining the techniques of latent variable analysis and SINDy, we infer key aspects of the interaction dynamics of CAR T-cell populations and cancer. Importantly, we show how the model terms can be interpreted biologically in relation to different CAR T-cell functional responses, single or double CAR T-cell-cancer cell binding models, and density-dependent growth dynamics in either of the CAR T-cell or cancer cell populations. We show how this data-driven model-discovery based approach provides unique insight into CAR T-cell dynamics when compared to an established model-first approach. These results demonstrate the potential for SINDy to improve the implementation and efficacy of CAR T-cell therapy in the clinic through an improved understanding of CAR T-cell dynamics.

Bifurcation analysis of a mathematical model of microalgae growth under the influence of sunlight

In this paper is considered a microalgae growth model under the influence of sunlight. The model is a two-dimensional system of the first order Ordinary Differential Equations (ODE) with a ten-dimensional parameter space. For this model, we study the existence of equilibrium points and their stability, and determine a bifurcation of the system when the value of some parameters is varied. The Lambert w function is used to calculate equilibrium points and apply the linearization technique to provide their stabilities. By varying the value of some parameters numerically, we found a transcritical bifurcation of the system and show stability regions of the equilibrium points in parameter diagrams. The bifurcation shows that the microalgae have a minimum sustainable nutrition supply and have a minimum light intensity that plays an important role for survival in a chemostat which has a certain depth. The results can be used to design a chemostat in optimizing the growth of microalgae.

Toward a logistic model of dynamic mold growth on wood

This paper deals with mathematical modeling of mold growth on wood. The logistic growth equation is used to model mold growth phenomena in changing environmental conditions. Microscopic mold coverage fraction is used in the logistic equation to represent mold development on a material surface over time. The current state-of-the-art dynamic Finnish mold growth model is used as the reference model for development of the logistic mold growth model. The optimal values of mold growth coefficient are estimated by simulations of mold growth under constant environmental conditions and by finding the best agreement with the Finnish mold growth model using mathematical optimization methods. It is shown that the mold growth coefficient can be approximated as a function of relative humidity, temperature, wood species and the quality of the surface. The model parameters are then verified by a comparison with the Finnish mold growth model. In this case, cyclic step changes of constant relative humidity are used. Finally, the logistic mold growth model is compared with published experimental observations of mold growth. The discrepancy of mold growth models and experimental mold growth data, and possible improvements are discussed.

Modelling Radiation Cancer Treatment with a Death-Rate Term in Ordinary and Fractional Differential Equations

Fractional calculus has recently been applied to the mathematical modelling of tumour growth, but its use introduces complexities that may not be warranted. Mathematical modelling with differential equations is a standard approach to study and predict treatment outcomes for population-level and patient-specific responses. Here, we use patient data of radiation-treated tumours to discuss the benefits and limitations of introducing fractional derivatives into three standard models of tumour growth. The fractional derivative introduces a history-dependence into the growth function, which requires a continuous death-rate term for radiation treatment. This newly proposed radiation-induced death-rate term improves computational efficiency in both ordinary and fractional derivative models. This computational speed-up will benefit common simulation tasks such as model parameterization and the construction and running of virtual clinical trials.

Predicting non-axisymmetric growth and facet evolution during lithiation of crystalline silicon anode particles through orientation-dependent interface reaction

A finite deformation modeling framework is developed to capture the experimentally observed two-phase, non-axisymmetric growth during the first lithiation of crystalline silicon cylinder in an improved fashion. The key feature of this model is an orientation-dependent interface reaction constant. Furthermore, the migration of lithium through the lithium-rich, growing amorphous zone is modeled through two additional features. First, an alloying–dealloying reaction that distinguishes between “movable” and “immovable” lithium parts. Second, a strong two-way coupling between diffusion and stress in contrast to existing mathematical models of non-axisymmetric growth. These features enable the prediction of a non-uniform distribution of concentration and stresses. The most important ability of this framework is that it can capture the evolution of facets at the crystalline–amorphous interface shown in experimental studies. The ensuing orientation-dependent interface velocity breaks the axisymmetric shape of the nanowire despite the initial circular geometry and the maintenance of an axisymmetric lithium influx. A very interesting finding is that the deposition of lithium and induced stresses are higher at the sites with lower interface velocity, which occur at the junction of two facets. The resulting stress concentration, in turn, pushes towards the further accumulation of lithium at those sites. It is expected that the improved modeling capabilities of this framework will contribute a step in moving beyond the limitations of simplistic structural design of lithium-ion battery particles.

Input significance ranking of microalgae continuous culture models

AbstractBackgroundMicroalgal cultures are evolving into a promising ecofriendly technology for a host of applications. To be sustainable, culture conditions need to be optimized and then controlled. One way to develop robust controllers for a cultivation system is by using mathematical growth models to simulate microalga‐based production. In this scenario, engineering design tasks begin by selecting the critical variables of these models.ResultsA new methodology for determining the significance ranking of a model's input factors under steady‐state operation (parameters (e.g. biological, geometrical) and/or process variables) was designed. The sensitivity of biomass response to its inputs was investigated in four different photobioreactor growth models within a nominal operational region. The methodology ranks models’ input factors based on the one‐at‐a‐time Morris method of elementary effects and variance‐based Sobol's method. Such information provided by the presented procedure is valuable as it reveals which input parameters explain most of the variance in model predictions.ConclusionThe methodology allowed the identification of controlled variables and biological parameters to be targeted for enhanced calibration. Furthermore, the presented methodology showed that in continuous reactors the dilution rate is a critical variable of the process. Therefore, it should be controlled. Additionally, most surprisingly, it is observed that controlling the light intensity within the optimum point of operation is not necessarily a crucial task. However, although its manipulation is still important, the accurate calibration of the parameters of the model may represent a greater influence on the biomass response. © 2023 Society of Chemical Industry (SCI).

Quantifying the Morphology and Mechanisms of Cancer Progression in 3D In-Vitro Environments: Integrating Experiments and Multiscale Models.

Mathematical models of cancer growth have become increasingly more accurate both in the space and time domains. However, the limited amount of data typically available has resulted in a larger number of qualitative rather than quantitative studies. In the present study, we provide an integrated experimental-computational framework for the quantification of the morphological characteristics and the mechanistic modelling of cancer progression in 3D environments. The proposed framework allows for the calibration of multiscale, spatiotemporal models of cancer growth using state-of-the-art 3D cell culture data, and their validation based on the resulting experimental morphological patterns using spatial point-pattern analysis techniques. We applied this framework to the study of the development of Triple Negative Breast Cancer cells cultured in Matrigel scaffolds, and validated the hypothesis of chemotactic migration using a multiscale, hybrid Keller-Segel model. The results revealed transient, non-random spatial distributions of cancer cells that consist of clustered, and dispersion patterns. The proposed model was able to describe the general characteristics of the experimental observations and suggests that chemotactic migration together with random motion was found to be a plausible mechanism leading to accumulation, during the examined time period of development. The developed framework enabled us to pursue two goals; first, the quantitative description of the morphology of cancer growth in 3D cultures using point-pattern analysis, and second, the relation of tumour morphology with underlying biophysical mechanisms that govern cancer growth and migration.

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.

Optimal Regulation Strategy for Nonzero-Sum Games of the Immune System Using Adaptive Dynamic Programming.

This article investigates the optimal control strategy problem for nonzero-sum games of the immune system based on adaptive dynamic programming (ADP). First, the main objective is approximating a Nash equilibrium between the tumor cells and the immune cell population, which is governed through chemotherapy drugs and immunoagents guided by the mathematical growth model of the tumor cells. Second, a novel intelligent nonzero-sum games-based ADP is put forward to solve the optimization control problem by reducing the growth rate of tumor cells and minimizing chemotherapy drugs and immunotherapy drugs. Meanwhile, the convergence analysis and iterative ADP algorithm are specified to prove feasibility. Finally, simulation examples are listed to account for availability and effectiveness of the research methodology.

Myeloid-Derived Suppressor-Cell Dynamics Control Outcomes in the Metastatic Niche.

Myeloid-derived suppressor cells (MDSC) play a prominent role in the tumor microenvironment. A quantitative understanding of the tumor-MDSC interactions that influence disease progression is critical, and currently lacking. We developed a mathematical model of metastatic growth and progression in immune-rich tumor microenvironments. We modeled the tumor-immune dynamics with stochastic delay differential equations and studied the impact of delays in MDSC activation/recruitment on tumor growth outcomes. In the lung environment, when the circulating level of MDSCs was low, the MDSC delay had a pronounced impact on the probability of new metastatic establishment: blocking MDSC recruitment could reduce the probability of metastasis by as much as 50%. To predict patient-specific MDSC responses, we fit to the model individual tumors treated with immune checkpoint inhibitors via Bayesian parameter inference. We reveal that control of the inhibition rate of natural killer (NK) cells by MDSCs had a larger influence on tumor outcomes than controlling the tumor growth rate directly. Posterior classification of tumor outcomes demonstrates that incorporating knowledge of the MDSC responses improved predictive accuracy from 63% to 82%. Investigation of the MDSC dynamics in an environment low in NK cells and abundant in cytotoxic T cells revealed, in contrast, that small MDSC delays no longer impacted metastatic growth dynamics. Our results illustrate the importance of MDSC dynamics in the tumor microenvironment overall and predict interventions promoting shifts toward less immune-suppressed states. We propose that there is a pressing need to consider MDSCs more often in analyses of tumor microenvironments.

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.

The Mathematical modeling of Cancer growth and angiogenesis by an individual based interacting system

We present a mathematical model for the complex system for the growth of a solid tumor. The system embeds proliferation of cells depending on the surrounding oxygen field, hypoxia caused by insufficient oxygen when the tumor reaches a certain size, consequent VEGF release and angiogenic new vasculature growth, re-oxygenation of the tumor and subsequent tumor growth restart. Specifically cancerous cells are represented by individual units, interacting as proliferating particles of a solid body, oxygen, and VEGF are fields with a source and a sink, and new angiogenic vasculature is described by a network of growing curves. The model, as shown by numerical simulations, captures both the time-evolution of the tumor growth before and after angiogenesis and its spatial properties, with different distribution of proliferating and hypoxic cells in the external and deep layers of the tumor, and the spatial structure of the angiogenic network. The microscopic description of the growth opens the possibility of tuning the model to patient-specific scenarios.

Integration of a System Dynamics Model and 3D Tree Rendering—VISmaF Part II: Model Development, Results and Potential Agronomic Applications

Biological–mathematical models of trees can be exploited for a wide range of agronomic applications including crop management, visualization of ecosystem changes over time, in-field phenotyping, crop load effects, testing of plant functions, biomechanics, and many others. Some models propose a 3D output of tree that, in addition to having functionality to visualize the result, offers an additional tool for the evaluation of some parameters of the model itself (interception and amount of light, temperature, obstacles, physical competition between multiple trees). The present study introduces a biological–mathematical model of tree growth with a 3D output of its structure in a realtime 3D rendering environment (Unity©). Thanks to the virtual environment created in Unity©, it was possible to obtain variable environmental parameters (amount of light, temperature) used as inputs to the mathematical simulation of growth. The model is based on ordinary differential equations (ODEs) that compute the growth of each single internode in length (primary growth) and width (secondary growth) and the accumulation of growth inhibitors regulating the seasonal cyclicity of the tree. Virtual experiments were conducted varying environmental conditions (amount of light and temperature), and the species-specific characteristics of the simulated tree (number of buds, branching angle). The results have been analyzed showing also how the model can be adapted for the creation of different tree species and discussing the potential agronomic applications of model.

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 Model of Avascular Tumor Growth: Necrosis and Apoptosis

The first section of this paper reviews the research on the development of avascular tumours, which are tumours without a blood artery network to supply their cells with nutrients. The process of avascular tumours is then illustrated utilising classical PDE model, which is composed of an outer layer of actively dividing cells and an inner core of dead cells (referred to as necrotic cells) (called proliferating cells). The model takes into account both necrosis (a type of cell death brought on by changes in the cell’s microenvironment) and apoptosis (a sort of programmed cell death that takes place when a cell reaches the end of its normal lifespan). We offer a modified, simplified model and study how the relative importance of these mechanisms shifts as the tumour enlarges using a mix of numerical and analytical methods.