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1
TANG, BETTY. "MODELING THE CELL DIVISION CYCLE: A QUALITATIVE APPROACH." Journal of Biological Systems 03, no. 01 (March 1995): 55–61. http://dx.doi.org/10.1142/s021833909500006x.
Повний текст джерелаАнотація:
The underlying biochemical mechanisms that drive the cell division cycle involve the interactions and feedback controls between the cytoplasmic proteins cdc2 and cyclin, and the activities of the cdc2-cyclin complex MPF. Alternation between interphase and mitosis is associated with oscillatory MPF and cyclin levels. This paper describes an ordinary differential equations (ODE) model and a functional differential equations (FDE) model of the cell cycle based on experimental work with the newly fertilized frog egg. One major difference of these models from previous ones is the use of nonspecific reaction terms in describing the interactions between cdc2, cyclin and MPF. This qualitative approach makes possible the evaluation of the roles of the various reactions and feedback mechanisms in the control of the cell cycle.
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Tyson, John J. "From the Belousov–Zhabotinsky reaction to biochemical clocks, traveling waves and cell cycle regulation." Biochemical Journal 479, no. 2 (January 28, 2022): 185–206. http://dx.doi.org/10.1042/bcj20210370.
Повний текст джерелаАнотація:
In the last 20 years, a growing army of systems biologists has employed quantitative experimental methods and theoretical tools of data analysis and mathematical modeling to unravel the molecular details of biological control systems with novel studies of biochemical clocks, cellular decision-making, and signaling networks in time and space. Few people know that one of the roots of this new paradigm in cell biology can be traced to a serendipitous discovery by an obscure Russian biochemist, Boris Belousov, who was studying the oxidation of citric acid. The story is told here from an historical perspective, tracing its meandering path through glycolytic oscillations, cAMP signaling, and frog egg development. The connections among these diverse themes are drawn out by simple mathematical models (nonlinear differential equations) that share common structures and properties.
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Lo Presti, Elena, Laura D’Orsi, and Andrea De Gaetano. "A Mathematical Model of In Vitro Cellular Uptake of Zoledronic Acid and Isopentenyl Pyrophosphate Accumulation." Pharmaceutics 14, no. 6 (June 14, 2022): 1262. http://dx.doi.org/10.3390/pharmaceutics14061262.
Повний текст джерелаАнотація:
The mevalonate pathway is an attractive target for many areas of research, such as autoimmune disorders, atherosclerosis, Alzheimer’s disease and cancer. Indeed, manipulating this pathway results in the alteration of malignant cell growth with promising therapeutic potential. There are several pharmacological options to block the mevalonate pathway in cancer cells, one of which is zoledronic acid (ZA) (an N-bisphosphonate (N-BP)), which inhibits the farnesyl pyrophosphate (FPP) synthase enzyme, inducing cell cycle arrest, apoptosis, inhibition of protein prenylation, and cholesterol reduction, as well as leading to the accumulation of isopentenyl pyrophosphate (IPP). We extrapolated the data based on two independently published papers that provide numerical data on the uptake of zoledronic acid (ZA) and the accumulation of IPP (Ag) and its isomer over time by using in vitro human cell line models. Two different mathematical models for IPP kinetics are proposed. The first model (Model 1) is a simpler ordinary differential equation (ODE) compartmental system composed of 3 equations with 10 parameters; the second model (Model 2) is a differential algebraic equation (DAE) system with 4 differential equations, 1 algebraic equation and 13 parameters incorporating the formation of the ZA+enzyme+Ag complex. Each of the two models aims to describe two different experimental situations (continuous and pulse experiments) with the same ZA kinetics. Both models fit the collected data very well. With Model 1, we obtained a prevision accumulation of IPP after 24 h of 169.6 pmol/mgprot/h with an IPP decreasing rate per (pmol/mgprot) of ZA (kXGZ) equal to 13.24/h. With Model 2, we have comprehensive kinetics of IPP upon ZA treatment. We calculate that the IPP concentration was equal to 141.6 pmol/mgprot/h with a decreasing rate/percentage of 0.051 (kXGU). The present study is the first to quantify the influence of ZA on the pharmacodynamics of IPP. While still incorporating a small number of parameters, Model 2 better represents the complexity of the biological behaviour for calculating the IPP produced in different situations, such as studies on γδ T cell-based immunotherapy. In the future, additional clinical studies are warranted to further evaluate and fine-tune dosing approaches.
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Smieja, Jaroslaw, Andrzej Swierniak, and Zdzislaw Duda. "Gradient Method for Finding Optimal Scheduling in Infinite Dimensional Models of Chemotherapy." Journal of Theoretical Medicine 3, no. 1 (2000): 25–36. http://dx.doi.org/10.1080/10273660008833062.
Повний текст джерелаАнотація:
One of the major obstacles against succesful chemotherapy of cancer is the emergence of resistance of cancer cells to cytotoxic agents. Applying optimal control theory to mathematical models of cell cycle dynamics can be a very efficient method to understand and, eventually, overcome this problem. Results that have been hitherto obtained have already helped to explain some observed phenomena, concerning dynamical properties of cancer populations. Because of recent progress in understanding the way in which chemotherapy affects cancer cells, new insights and more precise mathematical formulation of control problem, in the meaning of finding optimal chemotherapy, became possible. This, together with a progress in mathematical tools, has renewed hopes for improving chemotherapy protocols. In this paper we consider a population of neoplastic cells stratified into subpopulations of cells of different types. Due to the mutational event a sensitive cell can acquire a copy of the gene that makes it resistant to the agent. Likewise, the division of resistant cells can result in the change of the number of gene copies. We convert the model in the form of an infinite dimensional system of ordinary differential state equations discussed in our previous publications (see e.g. Swierniak etal., 1996b; Polariski etal., 1997; Swierniak etaL, 1998c), into the integro-differential form. It enables application of the necessary conditions of optimality given by the appropriate version of Pontryagin's maximum principle, e.g. (Gabasov and Kirilowa, 1971). The performance index which should be minimized combines the negative cumulated cytotoxic effect of the drug and the terminal population of both sensitive and resistant neoplastic cells. The linear form of the cost function and the bilinear form of the state equation result in a bang-bang optimal control law. To find the switching times we propose to use a special gradient algorithm developed similarly to the one applied in our previous papers to finite dimensional problems (Duda 1994; 1997).
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Лузянина, Т., та T. Luzyanina. "Численный бифуркационный анализ математических моделей с запаздыванием по времени с использованием пакета программ DDE-BIFTOOL". Mathematical Biology and Bioinformatics 12, № 2 (13 грудня 2017): 496–520. http://dx.doi.org/10.17537/2017.12.496.
Повний текст джерелаАнотація:
Mathematical modeling with delay differential equations (DDEs) is widely used for analysis and making predictions in various areas of the life sciences, e.g., population dynamics, epidemiology, immunology, physiology, neural networks. The time delays in these models take into account a dependence of the present state of the modeled system on its past history. The delay can be related to the duration of certain hidden processes like the stages of the life cycle, the time between infection of a cell and the production of new viruses, the duration of the infectious period, the immune period and so on. Due to an infinite-dimensional nature of DDEs, analytical studies of the corresponding mathematical models can only give limited results. Therefore, a numerical analysis is the major way to achieve both a qualitative and quantitative understanding of the model dynamics. A bifurcation analysis of a dynamical system is used to understand how solutions and their stability change as the parameters in the system vary. The package DDE-BIFTOOL is the first general-purpose package for bifurcation analysis of DDEs. This package can be used to compute and analyze the local stability of steady-state (equilibria) and periodic solutions of a given system as well as to study the dependence of these solutions on system parameters via continuation. Further one can compute and continue several local and global bifurcations: fold and Hopf bifurcations of steady states; folds, period doublings and torus bifurcations of periodic orbits; and connecting orbits between equilibria. In this paper we describe the structure of DDE-BIFTOOL, numerical methods implemented in the package and we illustrate the use of the package using a certain DDE system.
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Oduola, Wasiu Opeyemi, and Xiangfang Li. "Multiscale Tumor Modeling With Drug Pharmacokinetic and Pharmacodynamic Profile Using Stochastic Hybrid System." Cancer Informatics 17 (January 1, 2018): 117693511879026. http://dx.doi.org/10.1177/1176935118790262.
Повний текст джерелаАнотація:
Effective cancer treatment strategy requires an understanding of cancer behavior and development across multiple temporal and spatial scales. This has resulted into a growing interest in developing multiscale mathematical models that can simulate cancer growth, development, and response to drug treatments. This study thus investigates multiscale tumor modeling that integrates drug pharmacokinetic and pharmacodynamic (PK/PD) information using stochastic hybrid system modeling framework. Specifically, (1) pathways modeled by differential equations are adopted for gene regulations at the molecular level; (2) cellular automata (CA) model is proposed for the cellular and multicellular scales. Markov chains are used to model the cell behaviors by taking into account the gene expression levels, cell cycle, and the microenvironment. The proposed model enables the prediction of tumor growth under given molecular properties, microenvironment conditions, and drug PK/PD profile. Simulation results demonstrate the effectiveness of the proposed approach and the results agree with observed tumor behaviors.
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Craig, Morgan, Antony Humphries, Fahima Nekka, Jun Li, Jacques Bélair, and Michael Mackey. "Physiologically-Based Mathematical Modelling of Neutrophil Dynamics during Concurrent Chemotherapy and Filgrastim Support." Blood 124, no. 21 (December 6, 2014): 5134. http://dx.doi.org/10.1182/blood.v124.21.5134.5134.
Повний текст джерелаАнотація:
Abstract A common and dose-limiting side effect of chemotherapy is the development of neutropenia (a reduction in neutrophil numbers). To avoid or mitigate drops in absolute neutrophil counts (ANCs), patients are typically given recombinant human granulocyte colony-stimulating factor (rhG-CSF/filgrastim) to minimise the myelosuppressive nature of anti-cancer treatments. Dosing recommendations for filgrastim after chemotherapy suggest treatment begin one day post-chemotherapy and continue for a given amount of time or until ANCs rise sufficiently. Indeed, filgrastim support in a given chemotherapy cycle can sometimes reach seven to ten days in a 14-day period. Due to the intricacy of neutrophil production from the hematopoeitic stem cells in addition to the complexity of the interactions of cytokines and their receptors, a complete understanding of the mechanisms underlying myelopoiesis remains elusive. Mathematical modelling of these processes is a method which provides a global view of the dynamics of blood cell production and helps to elucidate the implications of concurrent chemotherapy and rhG-CSF support upon the blood production system. Moreover, the mathematical treatment of myelopoiesis can suggest novel dosing regimens that may be more beneficial than current schedules by supporting currently-held hypotheses and/or revealing previously unstudied relationships and dynamics. In this study, we construct a physiologically-based model of myelopoiesis which incorporates an up-to-date understanding of the production of neutrophils with our group's previously published model of blood cell dynamics. This model is combined with pharmacokinetic and pharmcodynamic (PKPD) models of Zalypsis (PM00104), an anti-cancer drug currently in phase II clinical trials, and filgrastim, a myelostimulant. The physiological model of myelopoiesis directly relates observable delays in neutrophil production to temporal lags in the model through the use of delay differential equations. All parameters are comprehensively defined for an average patient by utilising previously published physiological and PKPD studies. The model is numerically implemented and simulated to compare its predictions to ANC time series of patients undergoing the CHOP14 protocol. Able to recreate previously published data, we then investigated the optimal timing of filgrastim administrations post-chemotherapy during 14-day periodic chemotherapy and examined the number of filgrastim administrations necessary to ward off neutropenia using this optimised timing. Our results indicate that delaying rhG-CSF administrations by six or seven days after the administration of chemotherapy lessens the myelosuppressive impact of anti-cancer treatment. In addition, we found that if filgrastim administration are started seven days post-chemotherapy, as few as three or four doses of rhG-CSF during a 14-day cycle would improve the ANC nadir experienced by an average patient during myelosuppressive chemotherapy. In all, our results suggest that it is possible to lessen the hematopoietic burden of chemotherapy on patients and that detailed physiological modelling of myelopoiesis is a useful tool to clinicians and researchers alike. Disclosures Off Label Use: We look at optimal dosing regimens of filgrastim during periodic chemotherapy in the context of physiological mathematical models. No clinical trials were undertaken and no patients underwent any regimen changes..
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Neyhouse, Bertrand J., Jonathan Lee, and Fikile R. Brushett. "Predicting Cell Cycling Performance in Redox Flow Batteries Using Reduced-Order Analytical Models." ECS Meeting Abstracts MA2022-01, no. 3 (July 7, 2022): 474. http://dx.doi.org/10.1149/ma2022-013474mtgabs.
Повний текст джерелаАнотація:
Achieving decarbonization across multiple sectors (e.g., electricity generation, transportation, manufacturing) requires widespread adoption of renewable energy technologies, which demand energy storage solutions to enable sustainable, reliable, and resilient power delivery.1 To this end, redox flow batteries (RFBs) are a promising grid-scale energy storage platform, owing to their improved scalability, simplified manufacturing, and long service life.2 However, state-of-the-art RFBs remain too expensive for broad adoption, motivating the development of novel electrolyte formulations and reactor designs to meet performance, cost, and scale targets for emerging applications.3 While many recently-reported next-generation materials offer short-term performance improvements and the potential for cost reductions when produced at-scale, they often complicate system operation over extended durations due to a multitude of interrelated parasitic processes (e.g., side reactions, crossover, species decomposition) which lead to capacity fade and efficiency losses.3,4 Such processes challenge the establishment of quantitative and unambiguous connections between individual component properties and overall cell behavior. Here, we aim to develop mathematical models that translate fundamental material properties to cell performance metrics, enabling more informed design criteria for system engineering. In this presentation, we introduce an analytically-derived, zero-dimensional modeling framework to predict cell cycling behavior in RFBs. While previously-developed zero- and one-dimensional models demonstrate accurate performance predictions when compared to experimental systems, they must solve coupled differential equations using numerical methods.5,6 As a result, these approaches become computationally expensive for multi-cycle simulations (i.e., 10s – 100s of cycles), frustrating their implementation in system design and optimization. By deriving analytical solutions to these models, we can markedly reduce computation times and enable analyses hitherto unachievable. To demonstrate the utility of this modeling framework, we explore several representative scenarios that examine the connection between RFB material properties, operating conditions, and performance (i.e., power output, accessible capacity, efficiency). Additionally, we investigate the impact of different parasitic processes on capacity fade, highlighting the effects of species decomposition and crossover in durational cell cycling. Finally, we discuss several modalities for expanding this framework to include additional sources of performance losses and for integrating these models into larger computational schemes (e.g., optimization, parameter estimation, techno-economic assessments). The mathematical models developed in this work have potential to advance foundational understanding in RFB design, leading to quantitatively informed selection criteria for emerging candidate materials. Acknowledgments This work was supported by the Joint Center for Energy Storage Research, an Energy Innovation Hub funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences. B.J.N gratefully acknowledges the NSF Graduate Research Fellowship Program under Grant Number 1122374. J.L gratefully acknowledges support from the MIT Materials Research Laboratory REU Program, as part of the MRSEC Program of the NSF under grant number DMR-14-19807. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the NSF. References S. Chu and A. Majumdar, Nature, 488, 294–303 (2012). M. L. Perry and A. Z. Weber, J. Electrochem. Soc., 163, A5064–A5067 (2016). F. R. Brushett, M. J. Aziz, and K. E. Rodby, ACS Energy Lett., 5, 879–884 (2020). M. L. Perry, J. D. Saraidaridis, and R. M. Darling, Current Opinion in Electrochemistry, 21, 311–318 (2020). M. Pugach, M. Kondratenko, S. Briola, and A. Bischi, Applied Energy, 226, 560–569 (2018). S. Modak and D. G. Kwabi, J. Electrochem. Soc., 168, 080528 (2021).
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Miranda, Raquel, Susana Vinga, and Duarte Valério. "Studying Bone Remodelling and Tumour Growth for Therapy Predictive Control." Mathematics 8, no. 5 (May 1, 2020): 679. http://dx.doi.org/10.3390/math8050679.
Повний текст джерелаАнотація:
Bone remodelling consists of cycles of bone resorption and formation executed mainly by osteoclasts and osteoblasts. Healthy bone remodelling is disrupted by diseases such as Multiple Myeloma and bone metastatic diseases. In this paper, a simple mathematical model with differential equations, which takes into account the evolution of osteoclasts, osteoblasts, bone mass and bone metastasis growth, is improved with a pharmacokinetic and pharmacodynamic (PK/PD) scheme of the drugs denosumab, bisphosphonates, proteasome inhibitors and paclitaxel. The major novelty is the inclusion of drug resistance phenomena, which resulted in two variations of the model, corresponding to different paradigms of the origin and development of the tumourous cell resistance condition. These models are then used as basis for an optimization of the drug dose applied, paving the way for personalized medicine. A Nonlinear Model Predictive Control scheme is used, which takes advantage of the convenient properties of a suggested adaptive and democratic variant of Particle Swarm Optimization. Drug prescriptions obtained in this way provide useful insights into dose administration strategies. They also show how results may change depending on which of the two very different paradigms of drug resistance is used to model the behaviour of the tumour.
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Quiroga-Campano, Ana Luz, Louise Enfield, Matthew Foster, Margaritis Kostoglou, Michael Georgiadis, Athanasios Mantalaris, and Nicki Panoskaltsis. "Personalized and Optimized Low-Dose and Intensive Chemotherapy Treatments for Patients with Acute Myeloid Leukemia (AML)." Blood 132, Supplement 1 (November 29, 2018): 3500. http://dx.doi.org/10.1182/blood-2018-99-119258.
Повний текст джерелаАнотація:
Abstract Patients with AML have heterogeneous features, including those specific to the patient as well as those specific to the disease, such as leukemic burden and dynamic sub-clonal populations. Outside of clinical trials, few of these components are used to determine treatment. In order to move towards precision medicine, we have developed πiChemo, a computational application based on a dynamic mathematical modelling framework, using patient-, leukemia- and treatment-specific data to predict outcomes and optimize chemotherapy regimens for patients with AML. The model consists of a pharmacokinetic and pharmacodynamic (PK/PD) module that calculates the concentration and effect of Cytarabine Arabinoside (Ara-C) and Daunorubicin (DNR) in bone marrow (BM); and a population balance models (PBMs) module that describes normal populations (stem cells, progenitors, precursors) and abnormal populations (leukemic sensitive blasts (LSB) and leukemic resistant blasts (LRB)) in BM. The PBMs module also determines mature cell numbers in three lineages found in BM and peripheral blood (PB): (1) red blood cells (RBC), (2) white blood cells (WBC) and (3) lymphocytes (L). Model structure was analysed by global sensitivity analysis, which identified the most significant parameters on outcome predictions, re-estimated for each patient. The final integrated PK/PD & PBMs model has 1,295 differential equations, 8,044 algebraic equations, 9,335 variables, 25 fixed parameters and 4 degrees of freedom or variables to be optimized (Ara-C dose, Ara-C injection duration, DNR dose and DNR injection duration). Model validation, predictions and optimizations were performed using anonymised retrospective data from 28 patients with AML. The model required: (i) patient features: height, weight, age and gender, (ii) patient status: initial BM differential and PB cell counts, (iii) leukemia data: cellularity, presence of dysplasia and initial blast percentage and, (iv) treatment data: type (low-dose (LD) or intensive (DA)), dose, administration route (SC vs IV), administration mode (bolus injection vs infusion), time between injections and between cycles. The model predicted the absolute numbers of stem cells, progenitors, precursors, WBC, RBC, L, LSB and LRB in BM, and WBC, RBC, L and neutrophil count in PB during treatment for all patients. Model simulations predicted outcomes for 18 patients who achieved complete remission (7 LD & 11 DA), 4 patients who entered partial remission (2 LD & 2 DA) and 6 patients who relapsed (2 LD & 4 DA). The most remarkable results are those of prediction for BM blast percentage after each chemotherapy cycle and the PB neutrophil count for all patients. The notable fit between model predictions and daily patient data demonstrate model robustness and accuracy in the capacity to track patient-specific restaging BM and daily PB count evolution before, during and after treatment. The same patient datasets were used to apply an optimization algorithm that could maximize normal cell number and reduce leukemia burden, to personalize chemotherapy dose and administration for best outcomes. The results show that doses and administration methods vary between patients and between chemotherapy cycles for the same patient, depending on the evolution of normal and abnormal populations in BM. Low-dose continuous Ara-C infusions were more effective than rapid bolus injections, due to reduced chemotherapy effects on normal cells and subsequent quicker recovery in the normal BM compartments. RBC progenitors and precursors recovered faster than WBC and L lineages, and the recovery of normal BM cells was faster than that of normal mature cells in PB. The πiChemo tool requires only patient- and leukemia-specific initial conditions at diagnosis, easily obtained in standard clinical practice, for outcome predictions and treatment optimizations. Real-time model-fit testing and comparison of model results against daily PB cell counts would enable the re-estimation of significant parameters, increasing model accuracy and treatment effectiveness whilst therapy is ongoing. The πiChemo precision therapy tool has the potential to personalize optimal standard and novel treatments for AML in real-time. Disclosures No relevant conflicts of interest to declare.
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Pichor, Katarzyna, and Ryszard Rudnicki. "One and two-phase cell cycle models." BIOMATH 8, no. 1 (June 1, 2019): 1905261. http://dx.doi.org/10.11145/j.biomath.2019.05.261.
Повний текст джерелаАнотація:
In this review paper we present deterministic and stochastic one and two-phase models of the cell cycle. The deterministic models are given by partial differential equations of the first order with time delay and space variable retardation. The stochastic models are given by stochastic iterations or by piecewise deterministic Markov processes. We study asymptotic stability and sweeping of stochastic semigroups which describe the evolution of densities of these processes. We also present some results concerning chaotic behaviour of models and relations between different types of models.
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Zubik-Kowal, B. "Solutions for the Cell Cycle in Cell Lines Derived from Human Tumors." Computational and Mathematical Methods in Medicine 7, no. 4 (2006): 215–28. http://dx.doi.org/10.1080/10273660601017254.
Повний текст джерелаАнотація:
The goal of the paper is to compute efficiently solutions for model equations that have the potential to describe the growth of human tumor cells and their responses to radiotherapy or chemotherapy. The mathematical model involves four unknown functions of two independent variables: the time variabletand dimensionless relative DNA contentx. The unknown functions can be thought of as the number density of cells and are solutions of a system of four partial differential equations. We construct solutions of the system, which allow us to observe the number density of cells for differenttandxvalues. We present results of our experiments which simulate population kinetics of human cancer cellsin vitro. Our results show a correspondence between predicted and experimental data.
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Baker, C. T. H., G. A. Bocharov, and C. A. H. Paul. "Mathematical Modelling of the Interleukin-2 T-Cell System: A Comparative Study of Approaches Based on Ordinary and Delay Differential Equation." Journal of Theoretical Medicine 1, no. 2 (1997): 117–28. http://dx.doi.org/10.1080/10273669708833012.
Повний текст джерелаАнотація:
Cell proloferation and differentiation phenomena are key issues in immunology, tumour growth and cell biology. We study the kinetics of cell growth in the immune system using mathematical models formulated in terms of ordinary and delay differential equations. We study how the suitability of the mathematical models depends on the nature of the cell growth data and the types of differential equations by minimizing an objective function to give a best-fit parameterized solution. We show that mathematical models that incorporate a time-lag in the cell division phase are more consistent with certain reported data. They also allow various cell proliferation characteristics to be estimated directly, such as the average cell-boubling time and the rate of commitment of cells to cell division. Specifically, we study the interleukin-2-dependent cell division of phytohemagglutinin stimulated T-cells — the model of whic can be considered to be a general model of cell growth. We also review the numerical techniques available for solving delau differential equations and calculating the least-squares best-fit parameterized solution.
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Van der Hoff, Quay, and Ansie Harding. "Elemental access to limit cycle existence in Biomath education." Biomath Communications 6, no. 2 (January 10, 2020): 96. http://dx.doi.org/10.11145/bmc.2019.12.247.
Повний текст джерелаАнотація:
This paper originated from the desire to develop elementary calculus based tools to empower students, not necessarily with a strong mathematical background, to test predator-prey related models for boundedness of solutions and for the existence of limit cycles. There are several well-known methods available to prove, or disprove, the existence of bounded solutions to systems of differential equations. These methods rely on Liénard's theorem or using Dulac or Lyaponov functions. The level of mathematics required in the study of differential equations is not addressed in the courses presented on the first year level, and students in biology, ecology, economics and other fields are often not suitably equipped to perform these advanced techniques.The conditions under which a unique limit cycle exists in predator-prey systems is considered a primary problem in mathematical ecology. A great deal of mathematical effort has gone into trying to establish simple, yet general, theorems which will allow one to decide whether a given set of nonlinear equations has a limit cycle or not. We introduce a method to first determine the boundedness of solution trajectories in such a way that the transformation to a Liénard system or the use of a Dulac function can be avoided. Once boundedness of trajectories has been established, the nature of the equilibrium points reduces to simple eigenvalue analysis. The Elemental Limit Cycle method (ELC) provides elementary criteria to evaluate the nature of the pivotal functions of a system which will indicate boundedness and may be applicable to more general models.
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Apelinskiy, D. V., I. M. Shenderovskiy, and K. S. Runovskiy. "Mathematical problems of calculating the multi-zone models of a cycle of internal combustion engine with spark ignition." Izvestiya MGTU MAMI 7, no. 1-1 (January 10, 2013): 15–22. http://dx.doi.org/10.17816/2074-0530-68115.
Повний текст джерелаАнотація:
The article describes the features of mathematical models of internal combustion engine cycles which do not allow to put ordinary differential equations (ODE) of these models in the Cauchy form. The authors proposed a stable algorithm of numerical integration of ODE systems that are not solved with respect to derivatives, using Hermite splines and methods of nonlinear programming.
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Rangamani, Padmini, and Ravi Iyengar. "Modelling cellular signalling systems." Essays in Biochemistry 45 (September 30, 2008): 83–94. http://dx.doi.org/10.1042/bse0450083.
Повний текст джерелаАнотація:
Cell signalling pathways and networks are complex and often non-linear. Signalling pathways can be represented as systems of biochemical reactions that can be modelled using differential equations. Computational modelling of cell signalling pathways is emerging as a tool that facilitates mechanistic understanding of complex biological systems. Mathematical models are also used to generate predictions that may be tested experimentally. In the present chapter, the various steps involved in building models of cell signalling pathways are discussed. Depending on the nature of the process being modelled and the scale of the model, different mathematical formulations, ranging from stochastic representations to ordinary and partial differential equations are discussed. This is followed by a brief summary of some recent modelling successes and the state of future models.
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Ciliberto, Andrea, Bela Novak, and John J. Tyson. "Mathematical model of the morphogenesis checkpoint in budding yeast." Journal of Cell Biology 163, no. 6 (December 22, 2003): 1243–54. http://dx.doi.org/10.1083/jcb.200306139.
Повний текст джерелаАнотація:
The morphogenesis checkpoint in budding yeast delays progression through the cell cycle in response to stimuli that prevent bud formation. Central to the checkpoint mechanism is Swe1 kinase: normally inactive, its activation halts cell cycle progression in G2. We propose a molecular network for Swe1 control, based on published observations of budding yeast and analogous control signals in fission yeast. The proposed Swe1 network is merged with a model of cyclin-dependent kinase regulation, converted into a set of differential equations and studied by numerical simulation. The simulations accurately reproduce the phenotypes of a dozen checkpoint mutants. Among other predictions, the model attributes a new role to Hsl1, a kinase known to play a role in Swe1 degradation: Hsl1 must also be indirectly responsible for potent inhibition of Swe1 activity. The model supports the idea that the morphogenesis checkpoint, like other checkpoints, raises the cell size threshold for progression from one phase of the cell cycle to the next.
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Ahmed, Najma, Dumitru Vieru, and F. D. Zaman. "Memory effects on the proliferative function in the cycle-specific of chemotherapy." Mathematical Modelling of Natural Phenomena 16 (2021): 14. http://dx.doi.org/10.1051/mmnp/2021009.
Повний текст джерелаАнотація:
A generalized mathematical model of the breast and ovarian cancer is developed by considering the fractional differential equations with Caputo time-fractional derivatives. The use of the fractional model shows that the time-evolution of the proliferating cell mass, the quiescent cell mass, and the proliferative function are significantly influenced by their history. Even if the classical model, based on the derivative of integer order has been studied in many papers, its analytical solutions are presented in order to make the comparison between the classical model and the fractional model. Using the finite difference method, numerical schemes to the Caputo derivative operator and Riemann-Liouville fractional integral operator are obtained. Numerical solutions to the fractional differential equations of the generalized mathematical model are determined for the chemotherapy scheme based on the function of “on-off” type. Numerical results, obtained with the Mathcad software, are discussed and presented in graphical illustrations. The presence of the fractional order of the time-derivative as a parameter of solutions gives important information regarding the proliferative function, therefore, could give the possible rules for more efficient chemotherapy.
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Glagolev, Mikhail V., Aleksandr F. Sabrekov, and Vladimir M. Goncharov. "Delay differential equations as a tool for mathematical modelling of population dynamic." Environmental Dynamics and Global Climate Change 9, no. 2 (November 27, 2018): 40–63. http://dx.doi.org/10.17816/edgcc10483.
Повний текст джерелаАнотація:
The manuscript constitutes a lecture from a course “Mathematical modelling of biological processes”, adapted to the format of the journal paper. This course of lectures is held by one of authors in Ugra State University.
Delay differential equations are widely used in different ecological and biological problems. It has to do with the fact that delay differential equations are able to take into account that different biological processes depend not only on the state of the system at the moment but on the state of the system in previous moments too. The most popular case of using delay differential equations in biology is modelling in population ecology (including the modelling of several interacting populations dynamic, for example, in predator-prey system). Also delay differential equations are considered in demography, immunology, epidemiology, molecular biology (to provide mathematical description of regulatory mechanisms in a cell functioning and division), physiology as well as for modelling certain important production processes (for example, in agriculture).
In the beginning of the paper as introduction some basic concepts of differential difference equation theory (delay differential equations are specific type of differential difference equations) is considered and their classification is presented. Then it is discussed in more detail how such an important equations of population dynamic as Maltus equation and logistic (Verhulst-Pearl) equation are transformed into corresponsive delay differential equations – Goudriaan-Roermund and Hutchinson.
Then several discussion questions on using of a delay differential equations in biological models are considered. It is noted that in a certain cases using of a delay differential equations lead to an incorrect behavior from the point of view of common sense. Namely solution of Goudriaan-Roermund equation with harvesting, stopped when all species were harvested, shows that spontaneous generation takes place in the system.
This incorrect interpretation has to do with the fact that delay differential equations are used to simplify considered models (that are usually are systems of ordinary differential equations). Using of integro-differential equations could be more appropriate because in these equations background could be taken into account in a more natural way. It is shown that Hutchinson equation can be obtained by simplification of Volterra integral equation with a help of Diraq delta function.
Finally, a few questions of analytical and numerical solution of delay differential equations are discussed.
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Behaegel, Jonathan, Jean-Paul Comet, Gilles Bernot, Emilien Cornillon, and Franck Delaunay. "A hybrid model of cell cycle in mammals." Journal of Bioinformatics and Computational Biology 14, no. 01 (February 2016): 1640001. http://dx.doi.org/10.1142/s0219720016400011.
Повний текст джерелаАнотація:
Time plays an essential role in many biological systems, especially in cell cycle. Many models of biological systems rely on differential equations, but parameter identification is an obstacle to use differential frameworks. In this paper, we present a new hybrid modeling framework that extends René Thomas’ discrete modeling. The core idea is to associate with each qualitative state “celerities” allowing us to compute the time spent in each state. This hybrid framework is illustrated by building a 5-variable model of the mammalian cell cycle. Its parameters are determined by applying formal methods on the underlying discrete model and by constraining parameters using timing observations on the cell cycle. This first hybrid model presents the most important known behaviors of the cell cycle, including quiescent phase and endoreplication.
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Millan, Alberto Jovani, Suzanne Sindi, and Jennifer Manilay. "Predicting Natural Killer Cell Behavior with Mathematical Models." Journal of Immunology 202, no. 1_Supplement (May 1, 2019): 65.5. http://dx.doi.org/10.4049/jimmunol.202.supp.65.5.
Повний текст джерелаАнотація:
Abstract Natural killer (NK) cells are specialized lymphocytes with an innate ability to eliminate virally infected and cancerous cells, but the mechanisms that control NK cell development and cytotoxicity are incompletely understood. We hypothesize mathematical models can be used to predict how to direct NK cell development toward mature NK cells. We observed Sostdc1-knockout (KO) mice to display a partial NK cell developmental block. Building from these studies, a best fit model was simulated in which rates of transition and differentiation and cell death was determined by ordinary differential equations of deterministic compartmental models. This approach indicated that NK cell proliferation rates are not necessary to predict WT and KO population outcomes. We further hypothesize that clusters of highly cytotoxic NK cells can be identified computationally, using a combination of Ly49 receptor expression, in vitro and in vivo cytotoxicity assays using β2m−/− targets, and viSNE analysis of flow cytometry data. Sostdc1-KO NK cells also display defective cytotoxicity. We identified that KO splenic CD27+CD11b+ transitional NK cells express lower frequencies of inhibitory Ly49G2, but higher frequencies of activating Ly49H+ and D+ cells. Our viSNE results identified 3 NK cell clusters displaying a “superactive” phenotype that were virtually absent in KO mice; consistent with our observations of their hyporesponsiveness. Taken together, these data support a role for Sostdc1 in the regulation of NK cells and could provide insights into novel biological parameters to expand active NK cell numbers with high killing efficiency for immunotherapies.
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De Boeck, Jolan, Jan Rombouts, and Lendert Gelens. "A modular approach for modeling the cell cycle based on functional response curves." PLOS Computational Biology 17, no. 8 (August 11, 2021): e1009008. http://dx.doi.org/10.1371/journal.pcbi.1009008.
Повний текст джерелаАнотація:
Modeling biochemical reactions by means of differential equations often results in systems with a large number of variables and parameters. As this might complicate the interpretation and generalization of the obtained results, it is often desirable to reduce the complexity of the model. One way to accomplish this is by replacing the detailed reaction mechanisms of certain modules in the model by a mathematical expression that qualitatively describes the dynamical behavior of these modules. Such an approach has been widely adopted for ultrasensitive responses, for which underlying reaction mechanisms are often replaced by a single Hill function. Also time delays are usually accounted for by using an explicit delay in delay differential equations. In contrast, however, S-shaped response curves, which by definition have multiple output values for certain input values and are often encountered in bistable systems, are not easily modeled in such an explicit way. Here, we extend the classical Hill function into a mathematical expression that can be used to describe both ultrasensitive and S-shaped responses. We show how three ubiquitous modules (ultrasensitive responses, S-shaped responses and time delays) can be combined in different configurations and explore the dynamics of these systems. As an example, we apply our strategy to set up a model of the cell cycle consisting of multiple bistable switches, which can incorporate events such as DNA damage and coupling to the circadian clock in a phenomenological way.
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Cho, Heyrim, Ya-Huei Kuo, and Russell C. Rockne. "Comparison of cell state models derived from single-cell RNA sequencing data: graph versus multi-dimensional space." Mathematical Biosciences and Engineering 19, no. 8 (2022): 8505–36. http://dx.doi.org/10.3934/mbe.2022395.
Повний текст джерелаАнотація:
<abstract><p>Single-cell sequencing technologies have revolutionized molecular and cellular biology and stimulated the development of computational tools to analyze the data generated from these technology platforms. However, despite the recent explosion of computational analysis tools, relatively few mathematical models have been developed to utilize these data. Here we compare and contrast two cell state geometries for building mathematical models of cell state-transitions with single-cell RNA-sequencing data with hematopoeisis as a model system; (i) by using partial differential equations on a graph representing intermediate cell states between known cell types, and (ii) by using the equations on a multi-dimensional continuous cell state-space. As an application of our approach, we demonstrate how the calibrated models may be used to mathematically perturb normal hematopoeisis to simulate, predict, and study the emergence of novel cell states during the pathogenesis of acute myeloid leukemia. We particularly focus on comparing the strength and weakness of the graph model and multi-dimensional model.</p></abstract>
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Antonelli, P. L., S. F. Rutz, and V. S. Sabău. "A Transient-State Analysis of Tyson's Model for the Cell Division Cycle by Means of KCC-Theory." Open Systems & Information Dynamics 09, no. 03 (September 2002): 223–38. http://dx.doi.org/10.1023/a:1019752327311.
Повний текст джерелаАнотація:
The transient-state stability analysis for the trajectories of Tyson's equations for the cell-division cycle is given by the so-called KCC-Theory. This is the differential geometric theory of the variational equations for deviation of whole trajectories to nearby ones. The relationship between Lyapunov stability of steady-states and limit cycles is throughly examined. We show that the region of stability (where, in engineering parlance, the system is “hunting”) encloses the Tyson limit cycle, while outside this region the trajectories exhibit a periodic behaviour.
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Butler, George, Jonathan Rudge, and Philip R. Dash. "Mathematical modelling of cell migration." Essays in Biochemistry 63, no. 5 (October 2019): 631–37. http://dx.doi.org/10.1042/ebc20190020.
Повний текст джерелаАнотація:
Abstract The complexity of biological systems creates challenges for fully understanding their behaviour. This is particularly true for cell migration which requires the co-ordinated activity of hundreds of individual components within cells. Mathematical modelling can help understand these complex systems by breaking the system into discrete steps which can then be interrogated in silico. In this review, we highlight scenarios in cell migration where mathematical modelling can be applied and discuss what types of modelling are most suited. Almost any aspect of cell migration is amenable to mathematical modelling from the modelling of intracellular processes such as chemokine receptor signalling and actin filament branching to larger scale processes such as the movement of individual cells or populations of cells through their environment. Two common ways of approaching this modelling are the use of models based on differential equations or agent-based modelling. The application of both these approaches to cell migration are discussed with specific examples along with common software tools to facilitate the process for non-mathematicians. We also highlight the challenges of modelling cell migration and the need for rigorous experimental work to effectively parameterise a model.
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HYWOOD, JACK D., and KERRY A. LANDMAN. "BIASED RANDOM WALKS, PARTIAL DIFFERENTIAL EQUATIONS AND UPDATE SCHEMES." ANZIAM Journal 55, no. 2 (October 2013): 93–108. http://dx.doi.org/10.1017/s1446181113000369.
Повний текст джерелаАнотація:
AbstractThere is much interest within the mathematical biology and statistical physics community in converting stochastic agent-based models for random walkers into a partial differential equation description for the average agent density. Here a collection of noninteracting biased random walkers on a one-dimensional lattice is considered. The usual master equation approach requires that two continuum limits, involving three parameters, namely step length, time step and the random walk bias, approach zero in a specific way. We are interested in the case where the two limits are not consistent. New results are obtained using a Fokker–Planck equation and the results are highly dependent on the simulation update schemes. The theoretical results are confirmed with examples. These findings provide insight into the importance of updating schemes to an accurate macroscopic description of stochastic local movement rules in agent-based models when the lattice spacing represents a physical object such as cell diameter.
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HILLEN, T., K. J. PAINTER, and M. WINKLER. "Global solvability and explicit bounds for non-local adhesion models." European Journal of Applied Mathematics 29, no. 4 (November 23, 2017): 645–84. http://dx.doi.org/10.1017/s0956792517000328.
Повний текст джерелаАнотація:
Adhesion between cells and other cells (cell–cell adhesion) or other tissue components (cell–matrix adhesion) is an intrinsically non-local phenomenon. Consequently, a number of recently developed mathematical models for cell adhesion have taken the form of non-local partial differential equations, where the non-local term arises inside a spatial derivative. The mathematical properties of such a non-local gradient term are not yet well understood. Here we use sophisticated estimation techniques to show local and global existence of classical solutions for such examples of adhesion-type models, and we provide a uniform upper bound for the solutions. Further, we discuss the significance of these results to applications in cell sorting and in cancer invasion and support the theoretical results through numerical simulations.
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Fuentes-Garí, María, Ruth Misener, David García-Munzer, Eirini Velliou, Michael C. Georgiadis, Margaritis Kostoglou, Efstratios N. Pistikopoulos, Nicki Panoskaltsis, and Athanasios Mantalaris. "A mathematical model of subpopulation kinetics for the deconvolution of leukaemia heterogeneity." Journal of The Royal Society Interface 12, no. 108 (July 2015): 20150276. http://dx.doi.org/10.1098/rsif.2015.0276.
Повний текст джерелаАнотація:
Acute myeloid leukaemia is characterized by marked inter- and intra-patient heterogeneity, the identification of which is critical for the design of personalized treatments. Heterogeneity of leukaemic cells is determined by mutations which ultimately affect the cell cycle. We have developed and validated a biologically relevant, mathematical model of the cell cycle based on unique cell-cycle signatures, defined by duration of cell-cycle phases and cyclin profiles as determined by flow cytometry, for three leukaemia cell lines. The model was discretized for the different phases in their respective progress variables (cyclins and DNA), resulting in a set of time-dependent ordinary differential equations. Cell-cycle phase distribution and cyclin concentration profiles were validated against population chase experiments. Heterogeneity was simulated in culture by combining the three cell lines in a blinded experimental set-up. Based on individual kinetics, the model was capable of identifying and quantifying cellular heterogeneity. When supplying the initial conditions only, the model predicted future cell population dynamics and estimated the previous heterogeneous composition of cells. Identification of heterogeneous leukaemia clones at diagnosis and post-treatment using such a mathematical platform has the potential to predict multiple future outcomes in response to induction and consolidation chemotherapy as well as relapse kinetics.
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Matadi, Maba Boniface. "Application of Lie Symmetry to a Mathematical Model that Describes a Cancer Sub-Network." Symmetry 14, no. 2 (February 17, 2022): 400. http://dx.doi.org/10.3390/sym14020400.
Повний текст джерелаАнотація:
In this paper, a mathematical model of a cancer sub-network is analysed from the view point of Lie symmetry methods. This model discusses a human cancer cell which is developed due to the dysfunction of some genes at the R-checkpoint during the cell cycle. The primary purpose of this paper is to apply the techniques of Lie symmetry to the model and present some approximated solutions for the three-dimensional system of first-order ordinary differential equations describing a cancer sub-network. The result shows that the phosphatase gene (Cdc25A) regulates the cyclin-dependent kinases inhibitor (P27Kip1). Furthermore, this research discovered that the activity that reverses the inhibitory effects on cell cycle progression at the R-checkpoint initiates a pathway.
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Aubin-Frankowski, Pierre-Cyril, and Jean-Philippe Vert. "Gene regulation inference from single-cell RNA-seq data with linear differential equations and velocity inference." Bioinformatics 36, no. 18 (June 17, 2020): 4774–80. http://dx.doi.org/10.1093/bioinformatics/btaa576.
Повний текст джерелаАнотація:
Abstract Motivation Single-cell RNA sequencing (scRNA-seq) offers new possibilities to infer gene regulatory network (GRNs) for biological processes involving a notion of time, such as cell differentiation or cell cycles. It also raises many challenges due to the destructive measurements inherent to the technology. Results In this work, we propose a new method named GRISLI for de novo GRN inference from scRNA-seq data. GRISLI infers a velocity vector field in the space of scRNA-seq data from profiles of individual cells, and models the dynamics of cell trajectories with a linear ordinary differential equation to reconstruct the underlying GRN with a sparse regression procedure. We show on real data that GRISLI outperforms a recently proposed state-of-the-art method for GRN reconstruction from scRNA-seq data. Availability and implementation The MATLAB code of GRISLI is available at: https://github.com/PCAubin/GRISLI. Supplementary information Supplementary data are available at Bioinformatics online.
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Bernhardt, Knut. "Finding Alternatives and Reduced Formulations for Process-Based Models." Evolutionary Computation 16, no. 1 (March 2008): 63–88. http://dx.doi.org/10.1162/evco.2008.16.1.63.
Повний текст джерелаАнотація:
This paper addresses the problem of model complexity commonly arising in constructing and using process-based models with intricate interactions. Apart from complex process details the dynamic behavior of such systems is often limited to a discrete number of typical states. Thus, models reproducing the system's processes in all details are often too complex and over-parameterized. In order to reduce simulation times and to get a better impression of the important mechanisms, simplified formulations are desirable. In this work a data adaptive model reduction scheme that automatically builds simple models from complex ones is proposed. The method can be applied to the transformation and reduction of systems of ordinary differential equations. It consists of a multistep approach using a low dimensional projection of the model data followed by a Genetic Programming/Genetic Algorithm hybrid to evolve new model systems. As the resulting models again consist of differential equations, their process-based interpretation in terms of new state variables becomes possible. Transformations of two simple models with oscillatory dynamics, simulating a mathematical pendulum and predator-prey interactions respectively, serve as introductory examples of the method's application. The resulting equations of force indicate the predator-prey system's equivalence to a nonlinear oscillator. In contrast to the simple pendulum it contains driving and damping forces that produce a stable limit cycle.
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DE FEO, OSCAR. "QUALITATIVE RESONANCE OF SHIL'NIKOV-LIKE STRANGE ATTRACTORS, PART II: MATHEMATICAL ANALYSIS." International Journal of Bifurcation and Chaos 14, no. 03 (March 2004): 893–912. http://dx.doi.org/10.1142/s0218127404009739.
Повний текст джерелаАнотація:
This is the second of two papers introducing a new dynamical phenomenon, strongly related to the problems of synchronization and control of chaotic dynamical systems, and presenting the corresponding mathematical analysis, conducted both experimentally and theoretically. In particular, it is shown that different dynamical models (ordinary differential equations) admitting chaotic behavior organized by a homoclinic bifurcation to a saddle-focus (Shil'nikov-like chaos) tend to have a particular selective property when externally perturbed. Namely, these systems settle on a very narrow chaotic behavior, which is strongly correlated to the forcing signal, when they are slightly perturbed with an external signal which is similar to their corresponding generating cycle. Here, the "generating cycle" is understood to be the saddle cycle colliding with the equilibrium at the homoclinic bifurcation. On the other hand, when they are slightly perturbed with a generic signal, which has no particular correlation with their generating cycle, their chaotic behavior is reinforced. This peculiar behavior has been called qualitative resonance underlining the fact that such chaotic systems tend to resonate with signals that are qualitatively similar to an observable of their corresponding generating cycle. Here, a detailed mathematical analysis of the qualitative resonance phenomenon is presented, confirming the intuitions given by the geometrical model discussed in Part I.
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Reinharz, Vladimir, Alexander Churkin, Harel Dahari, and Danny Barash. "Advances in Parameter Estimation and Learning from Data for Mathematical Models of Hepatitis C Viral Kinetics." Mathematics 10, no. 12 (June 19, 2022): 2136. http://dx.doi.org/10.3390/math10122136.
Повний текст джерелаАнотація:
Mathematical models, some of which incorporate both intracellular and extracellular hepatitis C viral kinetics, have been advanced in recent years for studying HCV–host dynamics, antivirals mode of action, and their efficacy. The standard ordinary differential equation (ODE) hepatitis C virus (HCV) kinetic model keeps track of uninfected cells, infected cells, and free virus. In multiscale models, a fourth partial differential equation (PDE) accounts for the intracellular viral RNA (vRNA) kinetics in an infected cell. The PDE multiscale model is substantially more difficult to solve compared to the standard ODE model, with governing differential equations that are stiff. In previous contributions, we developed and implemented stable and efficient numerical methods for the multiscale model for both the solution of the model equations and parameter estimation. In this contribution, we perform sensitivity analysis on model parameters to gain insight into important properties and to ensure our numerical methods can be safely used for HCV viral dynamic simulations. Furthermore, we generate in-silico patients using the multiscale models to perform machine learning from the data, which enables us to remove HCV measurements on certain days and still be able to estimate meaningful observations with a sufficiently small error.
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Permyakova, A. V., A. V. Sazhin, E. V. Melekhina, and A. V. Gorelov. "POSSIBILITIES OF BIOLOGICAL AND MATHEMATICAL MODELING OF THE INFECTION CAUSED BY EPSTEIN–BARR VIRUS." Pediatria. Journal named after G.N. Speransky 99, no. 6 (December 14, 2020): 226–31. http://dx.doi.org/10.24110/0031-403x-2020-99-6-226-231.
Повний текст джерелаАнотація:
The review presents the existing biological and mathematical models of the infection process caused by the Epstein–Barr virus. The existence of the Epstein–Barr virus in the host organism can be described by a model representing a cycle of six consecutive stages, each of them has its own independent variant of immune regulation. The phenomenon of virus excretion in biological fluids, in particular, in saliva, is modeled using differential equations. Usage of mathematical modeling allows us to supplement existing knowledge about the pathogenesis of the infectious process caused by the Epstein–Barr virus, as well as to determine threshold levels of virus isolation in non-sterile environments for the diagnosis of active forms of infection.
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Volobuev, V., A. Kolsanov, N. Romanchuk, D. Romanov, I. Davydkin, and Pyatin. "Genetic-Mathematical Modeling of Population Interaction, New Psychoneuroimmunoendocrinology and Psychoneuroimmunology." Bulletin of Science and Practice 6, no. 11 (November 15, 2020): 85–103. http://dx.doi.org/10.33619/10.33619/2414-2948/60/09.
Повний текст джерелаАнотація:
Modern digital healthcare, biophysics and biology create new problems that stimulate the development of a new biophysical circuit and mathematical models from nuclear fusion (nuclear medicine) to genomic cell-organizational prognosis in neurophysiology, neuroendocrinology, psychoneuroimmunology and psychoneuroimmunoendocrinology. In this case, the following are effectively used: deterministic, stochastic, hybrid, multiscale modeling methods, as well as analytical and computational methods. The solution of a genetical-mathematical problem of interaction of cages of human population and virus population in relation to COVID-19 pandemic problem is submitted. A mathematical model based on the Hardy-Weinberg law is used, consisting of two interdependent differential equations. The equations reflect the temporal dynamics of the cells of human and viral populations during their interaction. Solutions of differential equations were found and the results of these solutions were analyzed. The pandemic duration is estimated using parameters of human liver cells and influenza virus. Perspective of further development of psychoneuroimmunology as interdisciplinary science, through algorithms and routing of digital health care, with expansion of psychoneurocommunications of professional interests in medicine, economics, sociology, cultural studies is shown. Modern neurobes and neuromarketing are built around Homo sapiens within a “reasonable environment” — a healthy individual space.
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Wattis, Jonathan A. D., Qi Qi, and Helen M. Byrne. "Mathematical modelling of telomere length dynamics." Journal of Mathematical Biology 80, no. 4 (November 14, 2019): 1039–76. http://dx.doi.org/10.1007/s00285-019-01448-y.
Повний текст джерелаАнотація:
AbstractTelomeres are repetitive DNA sequences located at the ends of chromosomes. During cell division, an incomplete copy of each chromosome’s DNA is made, causing telomeres to shorten on successive generations. When a threshold length is reached replication ceases and the cell becomes ‘senescent’. In this paper, we consider populations of telomeres and, from discrete models, we derive partial differential equations which describe how the distribution of telomere lengths evolves over many generations. We initially consider a population of cells each containing just a single telomere. We use continuum models to compare the effects of various mechanisms of telomere shortening and rates of cell division during normal ageing. For example, the rate (or probability) of cell replication may be fixed or it may decrease as the telomeres shorten. Furthermore, the length of telomere lost on each replication may be constant, or may decrease as the telomeres shorten. Where possible, explicit solutions for the evolution of the distribution of telomere lengths are presented. In other cases, expressions for the mean of the distribution are derived. We extend the models to describe cell populations in which each cell contains a distinct subpopulation of chromosomes. As for the simpler models, constant telomere shortening leads to a linear reduction in telomere length over time, whereas length-dependent shortening results in initially rapid telomere length reduction, slowing at later times. Our analysis also reveals that constant telomere loss leads to a Gaussian (normal) distribution of telomere lengths, whereas length-dependent loss leads to a log-normal distribution. We show that stochastic models, which include a replication probability, also lead to telomere length distributions which are skewed.
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Awang, Nor Aziran, Normah Maan, and Dasuki Sul’ain. "Tumour-Immune Interaction Model with Cell Cycle Effects Including G0 Phase." MATEMATIKA 34, no. 3 (December 31, 2018): 33–44. http://dx.doi.org/10.11113/matematika.v34.n3.1137.
Повний текст джерелаАнотація:
Tumour cells behave differently than normal cells in the body. They grow and divide in an uncontrolled manner (actively proliferating) and fail to respond to signal. However, there are cells that become inactive and reside in quiescent phase (G0). These cells are known as quiescence cells that are less sensitive to drug treatments (radiotherapy and chemotherapy) than actively proliferation cells. This paper proposes a new mathematical model that describes the interaction of tumour growth and immune response by considering tumour population that is divided into three different phases namely interphase, mitosis and G0. The model consists of a system of delay differential equations where the delay, represents the time for tumour cell to reside interphase before entering mitosis phase. Stability analysis of the equilibrium points of the system was performed to determine the dynamics behaviour of system. Result showed that the tumour population depends on number of tumour cells that enter active (interphase and mitosis) and G0phases. This study is important for treatment planning since tumour cell can resist treatment when they refuge in a quiescent state.
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FRIEDMAN, AVNER. "MATHEMATICAL ANALYSIS AND CHALLENGES ARISING FROM MODELS OF TUMOR GROWTH." Mathematical Models and Methods in Applied Sciences 17, supp01 (November 2007): 1751–72. http://dx.doi.org/10.1142/s0218202507002467.
Повний текст джерелаАнотація:
In the last four decades, various cancer models have been developed in which the evolution of the densities of cells (abnormal, normal, or dead) and the concentrations of biochemical species are described in terms of differential equations. In this paper, we deal with tumor models in which the tumor occupies a well-defined region in space; the boundary of this region is held together by the forces of cell-to-cell adhesion. We shall refer to such tumors as "solid" tumors, although they may sometimes consist of fluid-like tissue, such as in the case of brain tumors (e.g. gliomas) and breast tumors. The most common class of solid tumors is carcinoma: a cancer originating from epithelial cells, that is, from the closely packed cells which align the internal cavities of the body. Models of solid tumors must take spatial effects into account, and are therefore described in terms of partial differential equations (PDEs). They also need to take into account the fact that the tumor region is changing in time; in fact, the tumor region, say Ω(t), and its boundary Γ(t), are unknown in advance. Thus one needs to determine both the unknown "free boundary" Γ(t) together with the solution of the PDEs in Ω(t). These types of problems are called free boundary problems. The models described in this paper are free boundary problems, and our primary interest is the spatial/geometric features of the free boundary. Some of the basic questions we shall address are: What is the shape of the free boundary? How does the free boundary behave as t → ∞? Does the tumor volume increase or shrink as t → ∞? Under what conditions does the tumor eventually become dormant? Finally, we shall explore the dependence of the free boundary on some biological parameters, and this will give rise to interesting bifurcation phenomena. The structure of the paper is as follows. In Secs. 1 and 2 we consider models in which all the cells are of one type, they are all proliferating cells. The tissue is modeled either as a porous medium (in Sec. 1) or as a fluid medium (in Sec. 2). The models are extended in Secs. 3 and 4 to include three types of cells: proliferating, quiescent, and dead. Finally, in Sec. 5 we outline a general multiphase model that includes gene mutations.
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Anderson, A. R. A., M. A. J. Chaplain, E. L. Newman, R. J. C. Steele, and A. M. Thompson. "Mathematical Modelling of Tumour Invasion and Metastasis." Journal of Theoretical Medicine 2, no. 2 (2000): 129–54. http://dx.doi.org/10.1080/10273660008833042.
Повний текст джерелаАнотація:
In this paper we present two types of mathematical model which describe the invasion of host tissue by tumour cells. In the models, we focus on three key variables implicated in the invasion process, namely, tumour cells, host tissue (extracellular matrix) and matrix-degradative enzymes associated with the tumour cells. The first model focusses on the macro-scale structure (cell population level) and considers the tumour as a single mass. The mathematical model consists of a system of partial differential equations describing the production and/or activation of degradative enzymes by the tumour cells, the degradation of the matrix and the migratory response of the tumour cells. Numerical simulations are presented in one and two space dimensions and compared qualitatively with experimental and clinical observations. The second type of model focusses on the micro-scale (individual cell) level and uses a discrete technique developed in previous models of angiogenesis. This technique enables one to model migration and invasion at the level of individual cells and hence it is possible to examine the implications of metastatic spread. Finally, the results of the models are compared with actual clinical observations and the implications of the model for improved surgical treatment of patients are considered.
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Rajendran, Kylash, Irene M. Moroz, Scott M. Osprey, and Peter L. Read. "Descent Rate Models of the Synchronization of the Quasi-Biennial Oscillation by the Annual Cycle in Tropical Upwelling." Journal of the Atmospheric Sciences 75, no. 7 (June 25, 2018): 2281–97. http://dx.doi.org/10.1175/jas-d-17-0267.1.
Повний текст джерелаАнотація:
Abstract The response of the quasi-biennial oscillation (QBO) to an imposed mean upwelling with a periodic modulation is studied, by modeling the dynamics of the zero wind line at the equator using a class of equations known as descent rate models. These are simple mathematical models that capture the essence of QBO synchronization by focusing on the dynamics of the height of the zero wind line. A heuristic descent rate model for the zero wind line is described and is shown to capture many of the synchronization features seen in previous studies of the QBO. It is then demonstrated using a simple transformation that the standard Holton–Lindzen model of the QBO can itself be put into the form of a descent rate model if a quadratic velocity profile is assumed below the zero wind line. The resulting nonautonomous ordinary differential equation captures much of the synchronization behavior observed in the full Holton–Lindzen partial differential equation. The new class of models provides a novel framework within which to understand synchronization of the QBO, and we demonstrate a close relationship between these models and the circle map well known in the mathematics literature. Finally, we analyze reanalysis datasets to validate some of the predictions of our descent rate models and find statistically significant evidence for synchronization of the QBO that is consistent with model behavior.
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Sari, Eminugroho Ratna, Fajar Adi-Kusumo, and Lina Aryati. "Mathematical analysis of a SIPC age-structured model of cervical cancer." Mathematical Biosciences and Engineering 19, no. 6 (2022): 6013–39. http://dx.doi.org/10.3934/mbe.2022281.
Повний текст джерелаАнотація:
<abstract><p><italic>Human Papillomavirus</italic> (HPV), which is the main causal factor of cervical cancer, infects normal cervical cells on the specific cell's age interval, i.e., between the $ G_1 $ to $ S $ phase of cell cycle. Hence, the spread of the viruses in cervical tissue not only depends on the time, but also the cell age. By this fact, we introduce a new model that shows the spread of HPV infections on the cervical tissue by considering the age of cells and the time. The model is a four dimensional system of the first order partial differential equations with time and age independent variables, where the cells population is divided into four sub-populations, i.e., susceptible cells, infected cells by HPV, precancerous cells, and cancer cells. There are two types of the steady state solution of the system, i.e., disease-free and cancerous steady state solutions, where the stability is determined by using Fatou's lemma and solving some integral equations. In this case, we use a non-standard method to calculate the basic reproduction number of the system. Lastly, we use numerical simulations to show the dynamics of the age-structured system.</p></abstract>
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Sambono, Y., Zeth Arthur Leleury, Berny Pebo Tomasouw, and Dorteus L. Rahakbauw. "PENYELESAIAN SISTEM PEMBENTUKAN SEL PADA HYDRA MENGGUNAKAN METODE BEDA HINGGA SKEMA EKSPLISIT." BAREKENG: Jurnal Ilmu Matematika dan Terapan 14, no. 4 (December 1, 2020): 481–90. http://dx.doi.org/10.30598/barekengvol14iss4pp481-490.
Повний текст джерелаАнотація:
Mathematical models that describes the pattern of cell formation in hydra are expressed in a system of equations known as the Meinhardt model. This model is a continuous model in the form of diffusion equations. Thus, one of the studies which can be applied to Meinhardt equation is discretization. The finite difference model is a numerical method that can describe the discrete form of a continuous differential form. The method used in this study is finite different methods implementing explicit scheme. The advantage of the explicit scheme is easier to use for solving non-linear partial differential equations. This method used finite forward difference for derivatives of 𝑡 and finite centre difference for derivatives of 𝑥 at theactivator 𝑎(𝑥, 𝑡) and inhibitor 𝑏(𝑥, 𝑡). The Steps conducted by analyzing Meinhardt equation andcontinued with discretization such that earn the solution of system cell formation in hydra. According to the research its found that the activator cell population graphic have cell growth disposed ascend by the unit time, be different with the inhibitor cell population disposed descend of cell growth by the unit time.
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Michelson, Seth. "Multidrug Resistance and Its Reversal: Mathenatical Models." Journal of Theoretical Medicine 1, no. 2 (1997): 103–15. http://dx.doi.org/10.1080/10273669708833011.
Повний текст джерелаАнотація:
Classic multidrug resistance (MDR) is a phenomenon by which cells nonspecifically extrude noxious agents from the cutoplasm before lethal concentrations buils up. Some chemotherapeutically treated tumors exhibit these same dynamics. In tumor systems, the most common mechanism of facilitating MDR is the upregulation of the P-glycoprotein pump. This protein forms a transmembrance channel, and agter binding the chemotherapeutic agent and 2ATP molecules, forces the noxius agent through the channel. Hydrolysis of ATP to ADP provides the energy component of this reaction. General mathematical models describing drug resistamce are reviewed in this article. One model describing the molecular function of the P-glycoprotein pump in MDR cell lines is developed and presented in detail. The pump is modeled as an energy-dependent facilitated diffusion process. A partial differential equation is linked to a pair of ordinary differential equations to form the core of the model. To describe MDR reversal, the model is extended by additing an inhibitor to the equation system. Equations for competitive, one-site non-competitive, and allosteric non-competitive inhibition are then derived. Numerical simulations have been run to describe P-glycoprotein dynamics both in the presence and absence of inhibition, and these results are briefly reviewed. The character of the pump and its response to inhibition are discussed within the comtext of the models.
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Henriques Borges, Vanessa, Ivail Muniz Junior, Carlos Antonio De Moura, Dilson Silva, Celia Martins Cortez, and Maria Clicia Stelling de Castro. "Computational Mathematical Model Based on Lyapunov Function for the Hormonal Storage Control." International Journal for Innovation Education and Research 8, no. 11 (November 1, 2020): 375–91. http://dx.doi.org/10.31686/ijier.vol8.iss11.2761.
Повний текст джерелаАнотація:
Computational mathematical models have shown promise in the biological mechanism's reproduction. This work presents a computational mathematical model of the hormonal storage control applied to an endocrine cell. The model is based on a system of differential equations representing the internal cell dynamics and governed by the Lyapunov control function. Among the stages of these dynamics, we analyze the storage and degradation, which occur within some endocrine cells. The model’s evaluation considers, as an example, the synthesis–storage-release regulation of catecholamine in the adrenal medulla. Seven experiments, varying the input parameters, were performed to validate and evaluate the model. Different behaviors could be observed according to the numerical data used for future research and scientific contributions, besides confirming that Lyapunov control function is feasible to govern the cell dynamics.
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Tamilarasi, M., and R. Seyezhai. "State Space Averaged Modeling and Power Loss Computations for Fuel Cell Powered Four-Phase Interleaved Boost Converter." Advanced Materials Research 984-985 (July 2014): 1037–45. http://dx.doi.org/10.4028/www.scientific.net/amr.984-985.1037.
Повний текст джерелаАнотація:
This paper investigates the state space averaging of a four-phase Interleaved Boost Converter (IBC) powered by fuel cells. The state space averaging technique is a method to model the converter as time independent which is defined by a set of differential equations which will be useful for designing the controllers [18]. In this paper, mathematical models developed using state space averaging technique are presented for the proposed IBC. The power loss calculations are also computed. Simulation of IBC with fuel cell as the source is carried out in MATLAB and the results are presented and discussed.
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Orton, Richard J., Oliver E. Sturm, Vladislav Vyshemirsky, Muffy Calder, David R. Gilbert, and Walter Kolch. "Computational modelling of the receptor-tyrosine-kinase-activated MAPK pathway." Biochemical Journal 392, no. 2 (November 22, 2005): 249–61. http://dx.doi.org/10.1042/bj20050908.
Повний текст джерелаАнотація:
The MAPK (mitogen-activated protein kinase) pathway is one of the most important and intensively studied signalling pathways. It is at the heart of a molecular-signalling network that governs the growth, proliferation, differentiation and survival of many, if not all, cell types. It is de-regulated in various diseases, ranging from cancer to immunological, inflammatory and degenerative syndromes, and thus represents an important drug target. Over recent years, the computational or mathematical modelling of biological systems has become increasingly valuable, and there is now a wide variety of mathematical models of the MAPK pathway which have led to some novel insights and predictions as to how this system functions. In the present review we give an overview of the processes involved in modelling a biological system using the popular approach of ordinary differential equations. Focusing on the MAPK pathway, we introduce the features and functions of the pathway itself before comparing the available models and describing what new biological insights they have led to.
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Molina-Mora, José Arturo, Susana Mesen-Porras, Isaac Quiros-Fernandez, Mariana Kop-Montero, Andrea Rojas-Cespedes, Steve Quiros, Francisco Siles, and Rodrigo Mora. "Sphingolipid pathway as a biosensor of cancer chemosensitivity: a proof of principle." Uniciencia 36, no. 1 (November 1, 2022): 1–15. http://dx.doi.org/10.15359/ru.36-1.44.
Повний текст джерелаАнотація:
Cancer is a complex genetic disease with reduced treatment alternatives due to tumor heterogeneity and drug multiresistance emergence. The sphingolipid (SL) metabolic pathway integrates different responses of cellular stress signals and defines cell survival. Therefore, we suggest studying the perturbations on the sphingolipid pathway (SLP) caused by chemotherapeutic drugs using a systems biology approach to evaluate its functionality as a drug response sensor. We used a sphingomyelin-BODIPY (SM-BOD) sensor to study SL metabolism by flow cytometry and live cell imaging in different cancer models. To decode pathway complexity, we implemented Gussian mixture models, ordinary differential equations models, unsupervised classification algorithms and a fuzzy logic approach to assess its utility as a chemotherapy response sensor. Our results show that chemotherapeutic drugs perturb the SLP in different ways in a cell line-specific manner. In addition, we found that few SM-BOD fluorescence features predict chemosensitivity with high accuracy. Finally, we found that the relative species composition of SL appears to contribute to the resulting cytotoxicity of many treatments. This report offers a conceptual and mathematical framework for developing personalized mathematical models to predict and improve cancer therapy.
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SAVILL, N. J., and J. R. SEED. "Mathematical and statistical analysis of theTrypanosoma bruceislender to stumpy transition." Parasitology 128, no. 1 (January 2004): 53–67. http://dx.doi.org/10.1017/s0031182003004256.
Повний текст джерелаАнотація:
We propose a new model for the Stumpy Induction Factor-induced slender to stumpy transformation ofTrypanosoma brucei gambiensecells in immunosuppressed mice. The model is a set of delay differential equations that describe the time-course of the infection. We fit the model, using a maximum-likelihood method, to previously published data on parasitaemia in four mice. The model is shown to be a good fit and parameter estimates and confidence intervals are derived. Our estimated parameter values are consistent with estimates from previous experimental studies. The model predicts the following. Slender cells can be classified as uncommitted, committed and dividing, and committed and non-dividing. A committed slender cell undergoes about 5 divisions before exiting the cell-cycle. Committed slender cells must produce SIF, and stumpy cells must not produce SIF. There are two mechanisms for differentiation, a background differentiation rate, and a SIF-concentration-dependent differentiation rate, which is proportional to SIF concentration. SIF has a half-life of about 1·4 h in mice. We also show, with suitable changes in the parameter values, that the model reflects behaviours seen in other host species and trypanosome strains.
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Sinha, Sweta, Paramjeet Singh, and Mehmet Emir Koksal. "Mathematical and Numerical Modelling of Interference of Immune Cells in the Tumour Environment." Discrete Dynamics in Nature and Society 2023 (January 3, 2023): 1–18. http://dx.doi.org/10.1155/2023/9006678.
Повний текст джерелаАнотація:
In this article, the behaviour of tumour growth and its interaction with the immune system have been studied using a mathematical model in the form of partial differential equations. However, the development of tumours and how they interact with the immune system make up an extremely complex and little-understood system. A new mathematical model has been proposed to gain insight into the role of immune response in the tumour microenvironment when no treatment is applied. The resulting model is a set of partial differential equations made up of four variables: the population density of tumour cells, two different types of immune cells (CD4+ helper T cells and CD8+ cytotoxic T cells), and nutrition content. Such kinds of systems also occur frequently in science and engineering. The interaction of tumour and immune cells is exemplified by predator-prey models in ecology, in which tumour cells act as prey and immune cells act as predators. The tumour-immune cell interaction is expressed via Holling’s Type-III and Beddington-DeAngelis functional responses. The combination of finite volume and finite element method is used to approximate the system numerically because these approximations are more suitable for time-dependent systems having diffusion. Finally, numerical simulations show that the methods perform well and depict the behaviour of the model.
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Buzdugă, Ştefania Roxana, and Tudor Sajin. "Aspects of Mathematical Modelling of Heat and Mass Exchange at Burning of Droplet of Water Fuel Oil Emulsion." Applied Mechanics and Materials 371 (August 2013): 642–46. http://dx.doi.org/10.4028/www.scientific.net/amm.371.642.
Повний текст джерелаАнотація:
This paper presents some aspects regarding modelling of heat and mass exchange at burning of droplet of water-oil emulsion (WOE). Based on nonstationary differential equations of conservation of energy and matter in the liquid and gaseous phases, it was formulated a physical and mathematical model of heat and mass exchange in system of WOE drop-elementary gas cell. Unlike other models, in this model is taken into account the dependence of thermo physical properties of phase by temperature. The heating, micro explosion, vaporization and combustion stages of droplet burning were considered. For numerical solution and computer simulation of mathematical model has been developed an algorithm in Mathcad14 software. The numerical results were validated experimentally.