Bulletin of Mathematical Biology最新文献

Epidemiological Dynamics in Populations Structured by Neighbourhoods and Households.

IF 2 4区数学

Bulletin of Mathematical Biology Pub Date : 2025-02-27 DOI: 10.1007/s11538-025-01426-0

Abby Barlow, Sarah Penington, Ben Adams

{"title":"Epidemiological Dynamics in Populations Structured by Neighbourhoods and Households.","authors":"Abby Barlow, Sarah Penington, Ben Adams","doi":"10.1007/s11538-025-01426-0","DOIUrl":"10.1007/s11538-025-01426-0","url":null,"abstract":"Epidemiological dynamics are affected by the spatial and demographic structure of the host population. Households and neighbourhoods are known to be important groupings but little is known about the epidemiological interplay between them. In order to explore the implications for infectious disease epidemiology of households with similar demographic structures clustered in space we develop a multi-scale epidemic model consisting of neighbourhoods of households. In our analysis we focus on key parameters which control household size, the importance of transmission within households relative to outside of them, and the degree to which the non-household transmission is localised within neighbourhoods. We construct the household reproduction number <math><mmultiscripts><mi>R</mi> <mrow><mrow></mrow> <mo>∗</mo></mrow> <mrow></mrow></mmultiscripts> </math> over all neighbourhoods and derive the analytic probability of an outbreak occurring from a single infected individual in a specific neighbourhood. We find that reduced localisation of transmission within neighbourhoods reduces <math><mmultiscripts><mi>R</mi> <mrow><mrow></mrow> <mo>∗</mo></mrow> <mrow></mrow></mmultiscripts> </math> when household size differs between neighbourhoods. This effect is amplified by larger differences between household sizes and larger divergence between transmission rates within households and outside of them. However, the impact of neighbourhoods with larger household sizes on an individual's risk of infection is mainly limited to the individuals that reside in those neighbourhoods. We consider various surveillance scenarios and show that household size information from the initial infectious cases is often more important than neighbourhood information while household size and neighbourhood localisation influences the sequence of neighbourhoods in which an outbreak is observed.","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"87 4","pages":"50"},"PeriodicalIF":2.0,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143522736","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

An Asymptotic Analysis of Spike Self-Replication and Spike Nucleation of Reaction-Diffusion Patterns on Growing 1-D Domains. 增长的一维域上反应-扩散模式的尖峰自复制和尖峰成核的渐近分析。

IF 2 4区数学

Bulletin of Mathematical Biology Pub Date : 2025-02-24 DOI: 10.1007/s11538-025-01418-0

Chunyi Gai, Edgardo Villar-Sepúlveda, Alan Champneys, Michael J Ward

{"title":"An Asymptotic Analysis of Spike Self-Replication and Spike Nucleation of Reaction-Diffusion Patterns on Growing 1-D Domains.","authors":"Chunyi Gai, Edgardo Villar-Sepúlveda, Alan Champneys, Michael J Ward","doi":"10.1007/s11538-025-01418-0","DOIUrl":"https://doi.org/10.1007/s11538-025-01418-0","url":null,"abstract":"In the asymptotic limit of a large diffusivity ratio, certain two-component reaction-diffusion (RD) systems can admit localized spike solutions on a one-dimensional finite domain in a far-from-equilibrium nonlinear regime. It is known that two distinct bifurcation mechanisms can occur which generate spike patterns of increased spatial complexity as the domain half-length L slowly increases; so-called spike nucleation and spike self-replication. Self-replication is found to occur via the passage beyond a saddle-node bifurcation point that can be predicted through linearization around the inner spike profile. In contrast, spike nucleation occurs through slow passage beyond the saddle-node of a nonlinear boundary-value problem defined in the outer region away from the core of a spike. Here, by treating L as a static parameter under the Lagrangian framework, precise conditions are established within the semi-strong interaction asymptotic regime to determine which occurs, conditions that are confirmed by numerical simulation and continuation. For the Schnakenberg and Brusselator RD models, phase diagrams in parameter space are derived that predict whether spike self-replication or spike nucleation will occur first as L is increased, or whether no such instability will occur. For the Gierer-Meinhardt model with a non-trivial activator background, spike nucleation is shown to be the only possible spike-generating mechanism. From time-dependent PDE numerical results on an exponentially slowly growing domain, it is shown that the analytical thresholds derived from the asymptotic theory accurately predict critical values of L where either spike self-replication or spike-nucleation will occur. The global bifurcation mechanism for transitions to patterns of increased spatial complexity is further elucidated by superimposing time-dependent PDE simulation results on the numerically computed solution branches of spike equilibria in which L is the primary bifurcation parameter.","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"87 4","pages":"48"},"PeriodicalIF":2.0,"publicationDate":"2025-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143481833","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

EAD Mechanisms in Hypertrophic Mouse Ventricular Myocytes: Insights from a Compartmentalized Mathematical Model.

IF 2 4区数学

Bulletin of Mathematical Biology Pub Date : 2025-02-24 DOI: 10.1007/s11538-025-01423-3

Dilmini Warnakulasooriya, Vladimir E Bondarenko

{"title":"EAD Mechanisms in Hypertrophic Mouse Ventricular Myocytes: Insights from a Compartmentalized Mathematical Model.","authors":"Dilmini Warnakulasooriya, Vladimir E Bondarenko","doi":"10.1007/s11538-025-01423-3","DOIUrl":"https://doi.org/10.1007/s11538-025-01423-3","url":null,"abstract":"Transverse aortic constriction (TAC) is one of the experimental mouse models that are designed to investigate cardiac hypertrophy and heart failure. Most of the studies with this model are devoted to the stage of developed heart failure. However, several studies of the early stages (hypertrophy after 1 week of TAC) of this disease found significant changes in the β-adrenergic system, electrical activity, and Ca2+ dynamics in mouse ventricular myocytes. To provide a quantitative description of cardiac hypertrophy, we developed a new compartmentalized mathematical model of hypertrophic mouse ventricular myocytes for the early stage after the TAC procedure. The model described the changes in cell geometry, action potentials, [Ca2+]i transients, and β1- and β2-adrenergic signaling systems. We also showed that the hypertrophic myocytes demonstrated early afterdepolarizations (EADs) upon stimulation with isoproterenol at relatively long stimulation periods. Simulation of the hypertrophic myocyte activities revealed that the synergistic effects of the late Na+ current, the L-type Ca2+ current, and the T-type Ca2+ current were responsible for the initiation of EADs. The mechanisms of EAD and its suppression were investigated and sensitivity analysis was performed. Simulation results obtained with the hypertrophic cell model were compared to those from the normal ventricular myocytes. The developed mathematical model can be used for the explanation of the existing experimental data, for the development of the models for other hypertrophic phenotypes, and to make experimentally testable predictions of a hypertrophic myocyte's behavior.","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"87 4","pages":"49"},"PeriodicalIF":2.0,"publicationDate":"2025-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143481835","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

modelSSE: An R Package for Characterizing Infectious Disease Superspreading from Contact Tracing Data. modelSSE：从接触追踪数据描述传染病超级传播特征的 R 软件包。

IF 2 4区数学

Bulletin of Mathematical Biology Pub Date : 2025-02-21 DOI: 10.1007/s11538-025-01421-5

Shi Zhao, Zihao Guo, Kai Wang, Shengzhi Sun, Dayu Sun, Weiming Wang, Daihai He, Marc Kc Chong, Yuantao Hao, Eng-Kiong Yeoh

{"title":"modelSSE: An R Package for Characterizing Infectious Disease Superspreading from Contact Tracing Data.","authors":"Shi Zhao, Zihao Guo, Kai Wang, Shengzhi Sun, Dayu Sun, Weiming Wang, Daihai He, Marc Kc Chong, Yuantao Hao, Eng-Kiong Yeoh","doi":"10.1007/s11538-025-01421-5","DOIUrl":"https://doi.org/10.1007/s11538-025-01421-5","url":null,"abstract":"Infectious disease superspreading is a phenomenon where few primary cases generate unexpectedly large numbers of secondary cases. Superspreading, is frequently documented in epidemiology literature, and is considered a consequence of heterogeneity in transmission. Since understanding the risks of superspreading became a rising concern from both statistical modelling and public health aspects, the R package modelSSE provides comprehensive analytical tools to characterize transmission heterogeneity. The package modelSSE integrates recent advances in statistical methods, such as decomposition of reproduction number, for modelling infectious disease superspreading using various types and sources of contact tracing data that allow models to be grounded in real-world observations. This study provided an overview of the theoretical background and implementation of modelSSE, designed to facilitate learning infectious disease transmission, and explore novel research questions for transmission risks and superspreading potentials. Detailed examples of classic, historical infectious disease datasets are given for demonstration and model extensions.","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"87 4","pages":"47"},"PeriodicalIF":2.0,"publicationDate":"2025-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143467161","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Influence of Contact Lens Parameters on Tear Film Dynamics.

IF 2 4区数学

Bulletin of Mathematical Biology Pub Date : 2025-02-19 DOI: 10.1007/s11538-025-01425-1

Darshan Ramasubramanian, José Luis Hernández-Verdejo, José Manuel López-Alonso

{"title":"Influence of Contact Lens Parameters on Tear Film Dynamics.","authors":"Darshan Ramasubramanian, José Luis Hernández-Verdejo, José Manuel López-Alonso","doi":"10.1007/s11538-025-01425-1","DOIUrl":"https://doi.org/10.1007/s11538-025-01425-1","url":null,"abstract":"This study employs a computational model to simulate the dynamics of tear fluid and tear film in conjunction with contact lens motion, examining the interplay between diverse contact lens characteristics-such as material, design, and dimensions-and key ocular factors like dry eye conditions, corneal size, and blink rate. These interactions are critical for customising lens fit to maximise wearer comfort. Utilising optical measurements from a single participant, the study integrates data on tear meniscus size, blink velocity, and palpebral fissure height with sixteen different contact lens parameters, including Young's modulus, thickness, diameter, and curvature. Correlation analyses were conducted to determine the impact of these parameters on the dynamics of the tear fluid and overall tear film. Results show that the diameter and Young's modulus of the contact lens significantly influence pre-lens tear film thickness, with robust, statistically significant correlations. In contrast, lens thickness and base curve showed minimal impact, as evidenced by weak and non-significant correlations. These findings underscore the critical roles of lens diameter and Young's modulus in enhancing the stability and distribution of tear fluid, thereby improving wearer comfort and advancing contact lens design.","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"87 4","pages":"45"},"PeriodicalIF":2.0,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143448181","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Unveiling the Negative Synergistic Effect of Wall Shear Stress and Insulin on Endothelial NO Dynamics by Mathematical Modeling.

IF 2 4区数学

Bulletin of Mathematical Biology Pub Date : 2025-02-19 DOI: 10.1007/s11538-025-01424-2

Yu-Yuan Zhang, Yong-Jiang Li, Xu-Qu Hu, Chun-Dong Xue, Shen Li, Zheng-Nan Gao, Kai-Rong Qin

{"title":"Unveiling the Negative Synergistic Effect of Wall Shear Stress and Insulin on Endothelial NO Dynamics by Mathematical Modeling.","authors":"Yu-Yuan Zhang, Yong-Jiang Li, Xu-Qu Hu, Chun-Dong Xue, Shen Li, Zheng-Nan Gao, Kai-Rong Qin","doi":"10.1007/s11538-025-01424-2","DOIUrl":"https://doi.org/10.1007/s11538-025-01424-2","url":null,"abstract":"Diabetic vascular complications (DVCs) are diabetes-induced vascular dysfunction and pathologies, leading to the major causes of morbidity and mortality in millions of diabetic patients worldwide. DVCs are provoked by endothelial dysfunction which is closely coordinated with two important hallmarks: one is the insufficient insulin secretion or insulin resistance, and another is the decrease in intracellular nitric oxide (NO) influenced by dynamic wall shear stress (WSS). Although the intracellular NO dynamics in endothelial cells (ECs) is crucial for endothelial function, the regulation of NO production by dynamic WSS and insulin is still poorly understood. In this study, we have proposed a mathematical model of intracellular NO production in ECs under the stimulation of dynamic WSS combined with insulin. The model integrates simultaneously the biochemical signaling pathways of insulin and the mechanotransduction pathways induced by dynamic WSS. The accuracy and reliability of the model to quantitatively describe NO production in ECs were compared and validated with reported experimental data. According to the validated model, inhibition of protein kinase B (AKT) phosphorylation and Ca2+ influx by dynamic oscillatory WSS disrupts the dual nature of endothelial nitric oxide synthase (eNOS) enzyme activation. This disruption leads to the decrease in NO production and the bimodal disappearance of NO waveforms. Moreover, the results reveal that dynamic WSS combined with insulin promote endothelial NO production through negative synergistic effects, which is resulted from the temporal differences in mechanical and biochemical signaling. In brief, the proposed model elucidates the mechanism of NO generation activated by dynamic WSS combined with insulin, providing a potential target and theoretical framework for future treatment of DVCs.","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"87 4","pages":"46"},"PeriodicalIF":2.0,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143448184","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Correction to: Combining Mechanisms of Growth Arrest in Solid Tumours: A Mathematical Investigation.

IF 2 4区数学

Bulletin of Mathematical Biology Pub Date : 2025-02-12 DOI: 10.1007/s11538-024-01382-1

Chloé Colson, Helen M Byrne, Philip K Maini

引用次数: 0

A Simple Framework for Agent-Based Modeling with Extracellular Matrix. 基于细胞外基质的代理建模的简单框架

IF 2 4区数学

Bulletin of Mathematical Biology Pub Date : 2025-02-12 DOI: 10.1007/s11538-024-01408-8

John Metzcar, Ben S Duggan, Brandon Fischer, Matthew Murphy, Randy Heiland, Paul Macklin

{"title":"A Simple Framework for Agent-Based Modeling with Extracellular Matrix.","authors":"John Metzcar, Ben S Duggan, Brandon Fischer, Matthew Murphy, Randy Heiland, Paul Macklin","doi":"10.1007/s11538-024-01408-8","DOIUrl":"10.1007/s11538-024-01408-8","url":null,"abstract":"Extracellular matrix (ECM) is a key component of the cellular microenvironment and critical in multiple disease and developmental processes. Representing ECM and cell-ECM interactions is a challenging multiscale problem as they span molecular-level details to tissue-level dynamics. While several computational frameworks exist for ECM modeling, they often focus on very detailed modeling of individual ECM fibers or represent only a single aspect of the ECM. Using the PhysiCell agent-based modeling platform, we developed a framework of intermediate detail with the ability to capture bidirectional cell-ECM interactions. We represent a small region of ECM, an ECM element, with three variables describing its local microstructure: anisotropy, density, and overall fiber orientation. To spatially model the ECM, we use an array of ECM elements. Cells remodel local ECM microstructure and in turn, local microstructure impacts cellular motility. We demonstrate the utility of this framework and reusability of its core cell-ECM interaction model through examples in cellular invasion, wound healing, basement membrane degradation, and leader-follower collective migration. Despite the relative simplicity of the framework, it is able to capture a broad range of cell-ECM interactions of interest to the modeling community. Furthermore, variables representing the ECM microstructure are accessible through simple programming interfaces. This allows them to impact cell behaviors, such as proliferation and death, without requiring custom code for each interaction, particularly through PhysiCell's modeling grammar, enabling rapid modeling of a diverse range of cell-matrix biology. We make this framework available as a free and open source software package at https://github.com/PhysiCell-Models/collective-invasion .","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"87 3","pages":"43"},"PeriodicalIF":2.0,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11821717/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143397890","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Simplifying and Characterizing DAGs and Phylogenetic Networks via Least Common Ancestor Constraints.

IF 2 4区数学

Bulletin of Mathematical Biology Pub Date : 2025-02-12 DOI: 10.1007/s11538-025-01419-z

Anna Lindeberg, Marc Hellmuth

{"title":"Simplifying and Characterizing DAGs and Phylogenetic Networks via Least Common Ancestor Constraints.","authors":"Anna Lindeberg, Marc Hellmuth","doi":"10.1007/s11538-025-01419-z","DOIUrl":"10.1007/s11538-025-01419-z","url":null,"abstract":"Rooted phylogenetic networks, or more generally, directed acyclic graphs (DAGs), are widely used to model species or gene relationships that traditional rooted trees cannot fully capture, especially in the presence of reticulate processes or horizontal gene transfers. Such networks or DAGs are typically inferred from observable data (e.g., genomic sequences of extant species), providing only an estimate of the true evolutionary history. However, these inferred DAGs are often complex and difficult to interpret. In particular, many contain vertices that do not serve as least common ancestors (LCAs) for any subset of the underlying genes or species, thus may lack direct support from the observable data. In contrast, LCA vertices are witnessed by historical traces justifying their existence and thus represent ancestral states substantiated by the data. To reduce unnecessary complexity and eliminate unsupported vertices, we aim to simplify a DAG to retain only LCA vertices while preserving essential evolutionary information. In this paper, we characterize <math><mtext>LCA</mtext></math> -relevant and <math><mtext>lca</mtext></math> -relevant DAGs, defined as those in which every vertex serves as an LCA (or unique LCA) for some subset of taxa. We introduce methods to identify LCAs in DAGs and efficiently transform any DAG into an <math><mtext>LCA</mtext></math> -relevant or <math><mtext>lca</mtext></math> -relevant one while preserving key structural properties of the original DAG or network. This transformation is achieved using a simple operator \" <math><mo>⊖</mo></math> \" that mimics vertex suppression.","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"87 3","pages":"44"},"PeriodicalIF":2.0,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11821803/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143398056","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Effects of Non-linear Interaction Between Oxygen and Lactate on Solid Tumor Growth Under Cyclic Hypoxia.

IF 2 4区数学

Bulletin of Mathematical Biology Pub Date : 2025-02-11 DOI: 10.1007/s11538-025-01420-6

Gopinath Sadhu, D C Dalal

{"title":"Effects of Non-linear Interaction Between Oxygen and Lactate on Solid Tumor Growth Under Cyclic Hypoxia.","authors":"Gopinath Sadhu, D C Dalal","doi":"10.1007/s11538-025-01420-6","DOIUrl":"10.1007/s11538-025-01420-6","url":null,"abstract":"Oxygen is a crucial element for cellular respiration. Based on oxygen concentration, tumor regions can be categorized as normoxic, hypoxic, and necrotic. Hypoxic tumor cells switch their metabolism from aerobic glycolysis to anaerobic glycolysis. As a result, lactate is produced in hypoxic regions and is used as an alternative metabolic fuel by normoxic tumor cells. The consumption of lactate and oxygen by tumor cells does not follow a linear pattern. Scientific studies suggest that oxygen consumption and lactate production are non-linear phenomena. In this study, we propose a two-dimensional mathematical model to investigate lactate dynamics in avascular tumors with various initial shapes, such as circular, elliptical, and petal, and to explore its growth patterns in the context of non-linear interactions between oxygen and lactate. In certain human tumors, particularly in kidney, skin, and liver, multiple tumors may emerge within a tissue domain simultaneously. We also examine how the growth patterns of multiple tumors evolve within a shared domain. Cyclic hypoxia, commonly observed in solid tumors, results from oxygen fluctuations over time at the tumor site. Additionally, we analyze lactate dynamics and tumor growth patterns in environments with cyclic hypoxia. In order to simulate the proposed model, we use finite element based COMSOL Multiphysics 6.0 interface. The simulated results show excellent agreement with experimental data. Our findings reveal that the initial tumor shape significantly influences the lactate distribution and the tumor's internal structure. Furthermore, the simulations indicate that multiple tumors eventually merge into a single tumor. We also observe that cyclic hypoxia with short periodicity increases tumor volume.","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"87 3","pages":"41"},"PeriodicalIF":2.0,"publicationDate":"2025-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143398055","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0