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Bladder cancer poses a significant global health burden with high incidence and recurrence rates. This study addresses the therapeutic challenges in advanced bladder cancer, focusing on the competitive mechanisms of ligand or drug binding to receptors. We developed a refined mathematical model that integrates the dynamics of tumor cells and immune responses, particularly targeting fibroblast growth factor receptor 3 (FGFR3) and immune checkpoint inhibitors (ICIs). This study contributes to understanding combination therapies by elucidating the competitive binding dynamics and quantifying the synergistic effects. The findings highlight the importance of personalized immunotherapeutic strategies, considering factors such as drug dosage, dosing schedules, and patient-specific parameters. Our model further reveals that ligand-independent activated-state receptors are the most essential drivers of tumor proliferation. Moreover, we found that PD-L1 expression rate was more important than PD-1 in driving the dynamic evolution of tumor and immune cells. The proposed mathematical model provides a comprehensive framework for unraveling the complexities of combination therapies in advanced bladder cancer. As research progresses, this multidisciplinary approach contributes valuable insights toward optimizing therapeutic strategies and advancing cancer treatment paradigms.

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Patients with chronic myeloid leukemia (CML) who receive tyrosine kinase inhibitors (TKIs) have been known to achieve treatment-free remission (TFR) upon discontinuing treatment. However, the underlying mechanisms of this phenomenon remain incompletely understood. This study aims to elucidate the mechanism of TFR in CML patients, focusing on the feedback interaction between leukemia stem cells and the bone marrow microenvironment. We have developed a mathematical model to explore the interplay between leukemia stem cells and the bone marrow microenvironment, allowing for the simulation of CML progression dynamics. Our proposed model reveals a dichotomous response following TKI discontinuation, with two distinct patient groups emerging: one prone to early molecular relapse and the other capable of achieving long-term TFR after treatment cessation. This finding aligns with clinical observations and underscores the essential role of feedback interaction between leukemic cells and the tumor microenvironment in sustaining TFR. Notably, we have shown that the ratio of leukemia cells in peripheral blood (PBLC) and the tumor microenvironment (TME) index can be a valuable predictive tool for identifying patients likely to achieve TFR after discontinuing treatment. This study provides fresh insights into the mechanism of TFR in CML patients and underscores the significance of microenvironmental control in achieving TFR.

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Intratumoral heterogeneity and the presence of cancer stem cells are challenging issues in cancer therapy. An appropriate quantification of the stemness of individual cells for assessing the potential for self-renewal and differentiation from the cell of origin can define a measurement for quantifying different cell states, which is important in understanding the dynamics of cancer evolution, and might further provide possible targeted therapies aimed at tumor stem cells. Nevertheless, it is usually difficult to quantify the stemness of a cell based on molecular information associated with the cell. In this study, we proposed a stemness definition method with one-class Hadamard kernel support vector machine (OCHSVM) based on single-cell RNA sequencing (scRNA-seq) data. Applications of the proposed OCHSVM stemness are assessed by various data sets, including preimplantation embryo cells, induced pluripotent stem cells, or tumor cells. We further compared the OCHSVM model with state-of-the-art methods CytoTRACE, one-class logistic regression, or one-class SVM methods with different kernels. The computational results demonstrate that the OCHSVM method is more suitable for stemness identification using scRNA-seq data.

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Maintaining tissue homeostasis requires appropriate regulation of stem cell differentiation. The Waddington landscape posits that gene circuits in a cell form a potential landscape of different cell types, wherein cells follow attractors of the probability landscape to develop into distinct cell types. However, how adult stem cells achieve a delicate balance between self-renewal and differentiation remains unclear. We propose that random inheritance of epigenetic states plays a pivotal role in stem cell differentiation and present a hybrid model of stem cell differentiation induced by epigenetic modifications. Our comprehensive model integrates gene regulation networks, epigenetic state inheritance, and cell regeneration, encompassing multi-scale dynamics ranging from transcription regulation to cell population. Through model simulations, we demonstrate that random inheritance of epigenetic states during cell divisions can spontaneously induce cell differentiation, dedifferentiation, and transdifferentiation. Furthermore, we investigate the influences of interfering with epigenetic modifications and introducing additional transcription factors on the probabilities of dedifferentiation and transdifferentiation, revealing the underlying mechanism of cell reprogramming. This in silico model provides valuable insights into the intricate mechanism governing stem cell differentiation and cell reprogramming and offers a promising path to enhance the field of regenerative medicine.

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Excessive accumulation of ß-catenin proteins is a vital driver in the development of breast cancer. Many clinical assessments incorporating immunotherapy with targeted mRNA of ß-catenin are costly endeavor. This paper develops novel mathematical models for different treatments by invoking available clinical data to calibrate models, along with the selection and evaluation of therapy strategies in a faster manner with lower cost. Firstly, in order to explore the interactions between cancer cells and the immune system within the tumor microenvironment, we construct different types of breast cancer treatment models based on RNA interference technique and immune checkpoint inhibitors, which have been proved to be an effective combined therapy in pre-clinical trials associated with the inhibition of ß-catenin proteins to enhance intrinsic anti-tumor immune response. Secondly, various techniques including MCMC are adopted to estimate multiple parameters and thus simulations in agreement with experimental results sustain the validity of our models. Furthermore, the gradient descent method and particle swarm algorithm are designed to optimize therapy schemes to inhibit the growth of tumor and lower the treatment cost. Considering the mechanisms of drug resistance in vivo, simulations exhibit that therapies are ineffective resulting in cancer relapse in the prolonged time. For this reason, parametric sensitivity analysis sheds light on the choice of new treatments which indicate that, in addition to inhibiting ß-catenin proteins and improving self-immunity, the injection of dendritic cells promoting immunity may provide a novel vision for the future of cancer treatment. Overall, our study provides witness of principle from a mathematical perspective to guide clinical trials and the selection of treatment regimens.

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Dendritic cell (DC) vaccines and immune checkpoint inhibitors (ICIs) play critical roles in shaping the immune responses of tumor cells (TCs) and are widely used in cancer immunotherapies. Quantitatively evaluating the effectiveness of these therapies are essential for the optimization of treatment strategies. Here, based on the combined therapy of melanoma with DC vaccines and ICIs, we formulated a mathematical model to investigate the dynamic interactions between TCs and the immune system and understand the underlying mechanisms of immunotherapy. First, we obtained a threshold parameter for the growth of TCs, which is given by the ratio of spontaneous proliferation to immune inhibition. Next, we proved the existence and locally asymptotic stability of steady states of tumor-free, tumor-dominant, and tumor-immune coexistent equilibria, and identified the existence of Hopf bifurcation of the proposed model. Furthermore, global sensitivity analysis showed that the growth of TCs strongly correlates with the injection rate of DC vaccines, the activation rate of CTLs, and the killing rate of TCs. Finally, we tested the efficacy of multiple monotherapies and combined therapies with model simulations. Our results indicate that DC vaccines can decelerate the growth of TCs, and ICIs can inhibit the growth of TCs. Besides, both therapies can prolong the lifetime of patients, and the combined therapy of DC vaccines and ICIs can effectively eradicate TCs.

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The pandemic of novel coronavirus disease 2019 (COVID-19) has been a severe threat to public health. The policy of close contract tracing quarantine is an effective strategy in controlling the COVID-19 epidemic outbreak. In this paper, we developed a mathematical model of the COVID-19 epidemic with confirmed case-driven contact tracing quarantine, and applied the model to evaluate the effectiveness of the policy of contact tracing and quarantine. The model is established based on the combination of the compartmental model and individual-based model simulations, which results in a closed-form delay differential equation model. The proposed model includes a novel form of quarantine functions to represent the number of quarantine individuals following the confirmed cases every day and provides analytic expressions to study the effects of changing the quarantine rate. The proposed model can be applied to epidemic dynamics during the period of community spread and when the policy of confirmed cases-driven contact tracing quarantine is efficient. We applied the model to study the effectiveness of contact tracing and quarantine. The proposed delay differential equation model can describe the average epidemic dynamics of the stochastic-individual-based model, however, it is not enough to describe the diverse response due to the stochastic effect. Based on model simulations, we found that the policy of contact tracing and quarantine can obviously reduce the epidemic size, however, may not be enough to achieve zero-infectious in a short time, a combination of close contact quarantine and social contact restriction is required to achieve zero-infectious. Moreover, the effect of reducing epidemic size is insensitive to the period of quarantine, there are no significant changes in the epidemic dynamics when the quarantine days vary from 7 to 21 days.

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Although PD-1/PD-L1 inhibitors show potent and durable anti-tumour effects in some refractory tumours, the response rate in overall patients is unsatisfactory, which in part due to the inherent heterogeneity of PD-L1. In order to establish an approach for predicting and estimating the dynamic alternation of PD-L1 heterogeneity during cancer progression and treatment, this study establishes a comprehensive modelling and computational framework based on a mathematical model of cancer cell evolution in the tumour-immune microenvironment, and in combination with epigenetic data and overall survival data of clinical patients from The Cancer Genome Atlas. Through PD-L1 heterogeneous virtual patients obtained by the computational framework, we explore the adaptive therapy of administering anti-PD-L1 according to the dynamic of PD-L1 state among cancer cells. Our results show that in contrast to the continuous maximum tolerated dose treatment, adaptive therapy is more effective for PD-L1 positive patients, in that it prolongs the survival of patients by administration of drugs at lower dosage.

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Cancer is usually considered a genetic disease caused by alterations in genes that control cellular behaviors, especially growth and division. Cancer cells differ from normal tissue cells in many ways that allow them to grow out of control and become invasive. However, experiments have shown that aberrant growth in many tissues burdened with varying numbers of mutant cells can be corrected, and wild-type cells are required for the active elimination of mutant cells. These findings reveal the dynamic cellular behaviors that lead to a tissue homeostatic state when faced with mutational and nonmutational insults. The current study was motivated by these observations and established a mathematical model of how a tissue copes with the aberrant behavior of mutant cells. The proposed model depicts the interaction between wild-type and mutant cells through a system of two delay differential equations, which include the random mutation of normal cells and the active extrusion of mutant cells. Based on the proposed model, we performed qualitative analysis to identify the conditions of either normal tissue homeostasis or uncontrolled growth with varying numbers of abnormal mutant cells. Bifurcation analysis suggests the conditions of bistability with either a small or large number of mutant cells, the coexistence of bistable steady states can be clinically beneficial by driving the state of mutant cell predominance to the attraction basin of the state with a low number of mutant cells. This result is further confirmed by the treatment strategy obtained from optimal control theory.

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Intratumor heterogeneity hinders the success of anti-cancer treatment due to the interaction between different types of cells. To recapitulate the communication of different types of cells, we developed a mathematical model to study the dynamic interaction between normal, drug-sensitive and drug-resistant cells in response to cancer treatment. Based on the proposed model, we first study the analytical conclusions, namely the nonnegativity and boundedness of solutions, and the existence and stability of steady states. Furthermore, to investigate the optimal treatment that minimizes both the cancer cells count and the total dose of drugs, we apply the Pontryagin's maximum(or minimum) principle (PMP) to explore the combination therapy strategy with either quadratic control or linear control functionals. We establish the existence and uniqueness of the quadratic control problem, and apply the forward-backward sweep method (FBSM) to solve the optimal control problems and obtain the optimal therapy scheme.

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The COVID-19 (novel coronavirus disease 2019) pandemic has tremendously impacted global health and economics. Early detection of COVID-19 infections is important for patient treatment and for controlling the epidemic. However, many countries/regions suffer from a shortage of nucleic acid testing (NAT) due to either resource limitations or epidemic control measures. The exact number of infective cases is mostly unknown in counties/regions with insufficient NAT, which has been a major issue in predicting and controlling the epidemic. In this paper, we propose a mathematical model to quantitatively identify the influences of insufficient detection on the COVID-19 epidemic. We extend the classical SEIR (susceptible-exposed-infections-recovered) model to include random detections which are described by Poisson processes. We apply the model to the epidemic in Guam, Texas, the Virgin Islands, and Wyoming in the United States and determine the detection probabilities by fitting model simulations with the reported number of infected, recovered, and dead cases. We further study the effects of varying the detection probabilities and show that low level-detection probabilities significantly affect the epidemic; increasing the detection probability of asymptomatic infections can effectively reduce the the scale of the epidemic. This study suggests that early detection is important for the control of the COVID-19 epidemic.

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Patients with coronavirus disease 2019 (COVID-19) often exhibit diverse disease progressions associated with various infectious ability, symptoms, and clinical treatments. To systematically and thoroughly understand the heterogeneous progression of COVID-19, we developed a multi-scale computational model to quantitatively understand the heterogeneous progression of COVID-19 patients infected with severe acute respiratory syndrome (SARS)-like coronavirus (SARS-CoV-2). The model consists of intracellular viral dynamics, multicellular infection process, and immune responses, and was formulated using a combination of differential equations and stochastic modeling. By integrating multi-source clinical data with model analysis, we quantified individual heterogeneity using two indexes, i.e., the ratio of infected cells and incubation period. Specifically, our simulations revealed that increasing the host antiviral state or virus induced type I interferon (IFN) production rate can prolong the incubation period and postpone the transition from asymptomatic to symptomatic outcomes. We further identified the threshold dynamics of T cell exhaustion in the transition between mild-moderate and severe symptoms, and that patients with severe symptoms exhibited a lack of naïve T cells at a late stage. In addition, we quantified the efficacy of treating COVID-19 patients and investigated the effects of various therapeutic strategies. Simulations results suggested that single antiviral therapy is sufficient for moderate patients, while combination therapies and prevention of T cell exhaustion are needed for severe patients. These results highlight the critical roles of IFN and T cell responses in regulating the stage transition during COVID-19 progression. Our study reveals a quantitative relationship underpinning the heterogeneity of transition stage during COVID-19 progression and can provide a potential guidance for personalized therapy in COVID-19 patients.

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The outbreak of the novel coronavirus disease 2019 (COVID-19), caused by the novel severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), has caused global transmission, and been spread all over the world. For those regions that are currently free of infected cases, it is an urgent issue to prevent and control the local outbreak of COVID-19 when there are sporadic cases. To evaluate the effects of non-pharmaceutical interventions against local transmission of COVID-19, and to forecast the epidemic dynamics after local outbreak of diseases under different control measures, we developed an individual-based model (IBM) to simulate the transmission dynamics of COVID-19 from a microscopic perspective of individual-to-individual contacts to heterogenous among individuals. Based on the model, we simulated the effects of different levels of non-pharmaceutical interventions in controlling disease transmission after the appearance of sporadic cases. Simulations shown that isolation of infected cases and quarantine of close contacts alone would not eliminate the local transmission of COVID-19, and there is a risk of a second wave epidemics. Quarantine the second-layer close contacts can obviously reduce the size of outbreak. Moreover, to effectively eliminate the daily new infections in a short time, it is necessary to reduce the individual-to-individual contacts. IBM provides a numerical representation for the local transmission of infectious diseases, and extends the compartmental models to include individual heterogeneity and the close contacts network. Our study suggests that combinations of self-isolation, quarantine of close contacts, and social distancing would be necessary to block the local transmission of COVID-19.

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During mammalian embryo development, reprogramming of DNA methylation plays important roles in the erasure of parental epigenetic memory and the establishment of naive pluripotent cells. Multiple enzymes that regulate the processes of methylation and demethylation work together to shape the pattern of genome-scale DNA methylation and guide the process of cell differentiation. Recent availability of methylome information from single-cell whole genome bisulfite sequencing (scBS-seq) provides an opportunity to study DNA methylation dynamics in the whole genome in individual cells, which reveal the heterogeneous methylation distributions of enhancers in embryo stem cells. In this study, we developed a computational model of enhancer methylation inheritance to study the dynamics of genome-scale DNA methylation reprogramming during exit from pluripotency. The model enables us to track genome-scale DNA methylation reprogramming at single-cell level during the embryo development process and reproduce the DNA methylation heterogeneity reported by scBS-seq. Model simulations show that DNA methylation heterogeneity is an intrinsic property driven by cell division along the development process, and the collaboration between neighboring enhancers is required for heterogeneous methylation. Our study suggests that the mechanism of genome-scale oscillation might not be necessary for the DNA methylation heterogeneity during exit from pluripotency.

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Stem cell heterogeneity is essential for homeostasis in tissue development. This paper establishes a general mathematical framework to model the dynamics of stem cell regeneration with cell heterogeneity and random transitions of epigenetic states. The framework generalizes the classical G0 cell cycle model and incorporates the epigenetic states of individual cells represented by a continuous multidimensional variable. In the model, the kinetic rates of cell behaviors, including proliferation, differentiation, and apoptosis, are dependent on their epigenetic states, and the random transitions of epigenetic states between cell cycles are represented by an inheritance probability function that describes the conditional probability of cell state changes. Moreover, the model can be extended to include genotypic changes and describe the process of gene mutation-induced tumor development. The proposed mathematical framework provides a generalized formula that helps us to understand various dynamic processes of stem cell regeneration, including tissue development, degeneration, and abnormal growth.

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BACKGROUND: Most researches of chronic myeloid leukemia (CML) are currently focused on the treatment methods, while there are relatively few researches on the progress of patients' condition after drug treatment. Traditional biomarkers of disease can only distinguish normal state from disease state, and cannot recognize the pre-stable state after drug treatment. RESULTS: A therapeutic effect recognition strategy based on dynamic network biomarkers (DNB) is provided for CML patients' gene expression data. With the DNB criteria, the DNB with 250 genes is selected and the therapeutic effect index (TEI) is constructed for the detection of individual disease. The pre-stable state before the disease condition becomes stable is 1 month. Through functional analysis for the DNB, some genes are confirmed as key genes to affect the progress of CML patients' condition. CONCLUSIONS: The results provide a certain theoretical direction and theoretical basis for medical personnel in the treatment of CML patients, and find new therapeutic targets in the future. The biomarkers of CML can help patients to be treated promptly and minimize drug resistance, treatment failure and relapse, which reduce the mortality of CML significantly.

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In human tissues, the replicative potential of stem cells is limited by the shortening of telomere, limitless replicative potential is a hallmark of cancer. Telomere length changes stochastically during cell division mainly due to the competition between the end replication problem and telomerase, short telomere can lead to replicative senescence and cell apoptosis. Here, we investigate how stochastic changes of telomere length in individual cells may affect the population dynamics of clonal growth. We established a computational model that couples telomerase-regulated stochastic telomere length changes with the replicative potential of clones. Model simulations reveal qualitative dependence of clone proliferation potential with activities of telomerase; mutations in cells to alter the activities of telomerase and its inhibitors can induce abnormal tissue growth and lead to limitless replicative potential.

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Cyclical thrombocytopenia (CT) is a rare hematological disease characterized by periodic oscillations in circulating platelet counts. In almost all CT patients, other cell lines show no sign of oscillation, but recently a CT patient was reported with significant oscillations in circulating neutrophils (in the same period as the platelets). In this paper, we attempt to understand this phenomenon through a previously published model of human hematopoiesis. We have investigated a variety of possible oscillation patterns that may appear when alterations occur in the control parameters in the platelet regulatory dynamics. Our results indicate that the platelet maturation time and the differentiation rate from hematopoietic stem cells (HSCs) into the platelet cell line play important roles in the emergence of various types of CT like oscillations. Moreover, we find different oscillation patterns, including CT and cyclical neutropenia like oscillations, with certain parameter values in the platelet compartment. A bifurcation analysis revealed the different origins of these oscillation patterns. We also identified bistable dynamics which indicate the potential importance of system history in the treatment of these diseases. Together, these results demonstrate the possible origins for various oscillation patterns dependent on alterations in the platelet cell line control mechanisms. One of the important origins may be related to the regulation of apoptosis in platelet precursors.

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Coculture of mesenchymal stem cells (MSCs) and vascular endothelial cells (ECs) in vitro leads to the formation of a capillary-like reticular structure by ECs, which has great potential as a better substitute for artificial blood vessels in terms of stability and functionality. To investigate the mechanisms of the early neovascularization induced by MSCs, we analyzed the kinematic features of the motion of ECs and concluded that the dynamic interaction between cells and the extracellular matrix would reveal the capillary-like structure formation. Based on this hypothesis, we proposed a mathematical model to simulate the vascular-like migration pattern of ECs in silico, which was confirmed by in vitro studies. These in vitro studies validated that the dynamic secretion and degradation of collagen I is the critical factor for capillary structure formation. The model proposed based on cell tracking, single cell sequencing, and mathematical simulation provides a better understanding of the neovascularization process induced by MSCs and a possible simple explanation guiding this important cellular behavior.-Yu, Y., Situ, Q., Jia, W., Li, J., Wu, Q., Lei, J. Data driven mathematical modeling reveals the dynamic mechanism of MSC-induced neovascularization.

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