Mathematical modeling of combined therapies for treating tumor drug resistance.

Drug resistance is one of the most intractable issues to the targeted therapy for cancer diseases. To explore effective combination therapy schemes, we propose a mathematical model to study the effects of different treatment schemes on the dynamics of cancer cells. Then we characterize the dynamical behavior of the model by finding the equilibrium points and exploring their local stability. Lyapunov functions are constructed to investigate the global asymptotic stability of the model equilibria. Numerical simulations are carried out to verify the stability of equilibria and treatment outcomes using a set of collected model parameters and experimental data on murine colon carcinoma. Simulation results suggest that immunotherapy combined with chemotherapy contributes significantly to the control of tumor growth compared to monotherapy. Sensitivity analysis is performed to identify the importance of model parameters on the variations of model outcomes.

Mathematical modeling the gene mechanism of colorectal cancer and the effect of radiation exposure.

Cancer is the result of continuous accumulation of gene mutations in normal cells. The number of mutations is different in different types of cancer and even in different patients with the same type of cancer. Therefore, studying all possible numbers of gene mutations in malignant cells is of great value for the understanding of tumorigenesis and the treatment of cancer. To this end, we applied a stochastic mathematical model considering the clonal expansion of any premalignant cells with different mutations to analyze the number of gene mutations in colorectal cancer. The age-specific colorectal cancer incidence rates from the Surveillance, Epidemiology and End Results (SEER) registry in the United States and the Life Span Study (LSS) in Nagasaki and Hiroshima, Japan are chosen to test the reasonableness of the model. Our fitting results indicate that the transformation from normal cells to malignant cells may undergo two to five driver mutations for colorectal cancer patients without radiation-exposed environment, two to four driver mutations for colorectal cancer patients with low level radiation-exposure, and two to three driver mutations for colorectal cancer patients with high level radiation-exposure. Furthermore, the net growth rate of the mutated cells with radiation-exposure was is higher than that of the mutated cells without radiation-exposure for the models with two to five driver mutations. These results suggest that radiation environment may affect the clonal expansion of cells and significantly affect the development of tumors.

TSPLASSO: A Two-stage Prior LASSO Algorithm for Gene Selection using Omics Data.

Feature selection has been extensively applied to identify cancer genes using omics data. Although substantial studies have been conducted to search for cancer genes, the available rich knowledge on various cancers is seldom used as prior information in feature selection. This paper proposes a two-stage prior LASSO (TSPLASSO) method, which represents an early attempt in designing feature selection algorithms using prior information. The first stage performs gene selection via linear regression with LASSO. Candidate genes that are correlated with known cancer genes are retained for subsequent analysis. The second stage establishes a logistic regression model with LASSO to realize final cancer gene selection and sample classification. The key advantages of TSPLASSO include the successive consideration of prior cancer genes and binary sample types as response variables in stages one and two, respectively. In addition, the TSPLASSO performs sample classification and variable selection simultaneously. Compared with six state-of-the-art algorithms, numerical simulations in six real-world datasets show that TSPLASSO can improve the accuracy of variable selection by 5%-400% in the three bulk sequencing datasets and the scRNA-seq dataset; and the performance is robust against data noise and variations of prior cancer genes. The TSPLASSO provides an efficient, stable and practical algorithm for exploring biomedcial and health informatics from omics data.

Growth dynamics of breast cancer stem cells: effects of self-feedback and EMT mechanisms.

Breast cancer stem cells (BCSCs) with the ability to self-renew and differentiate have been identified in primary breast cancer tissues and cell lines. The BCSCs are often resistant to traditional radiation and/or chemotherapies. Previous studies have also shown that successful therapy must eradicate cancer stem cells. The purpose of this paper is to develop a mathematical model with self-feedback mechanism to illustrate the issues regarding the difficulties of absolutely eliminating a breast cancer. In addition, we introduce the mechanism of the epithelial-mesenchymal transition (EMT) to investigate the influence of EMT on the effects of breast cancer growth and treatment. Results indicate that the EMT mechanism facilitates the growth of breast cancer and makes breast cancer more difficult to be cured. Therefore, targeting the signals involved in EMT can halt tumor progression in breast cancer. Finally, we apply the experimental data to carry out numerical simulations and validate our theoretical conclusions.

Mathematical modeling for mutator phenotype and clonal selection advantage in the risk analysis of lung cancer.

Cancer is one of the leading diseases for human mortality. Although substantial research works have been conducted to investigate the initiation and progression of cancer disease, it is still an active debate regarding the function of mutations conferring a clone advantage and the importance of mutator phenotypes caused by the mutation of stability genes. To address this issue further, we develop a mathematical model based on the incidence data of non-small cell lung cancer and small cell lung cancer from the Surveillance Epidemiology and End Results registry in the USA. The key biological parameters have been analyzed to investigate the potential effective measures for inhibiting the risk of lung cancer. Although the first event is the gene mutation that leads to clonal expansion of cells for lung cancer, the simulation results show that the clonal advantage of cancer cells alone is insufficient to cause tumorigenesis. Our analysis suggests that mutations in genes that keep genetic stability are critical in the development of lung cancer. This implies that mutator phenotype is an important indicator for the diagnosis of lung cancer, which can enable early detection and treatment to reduce the risk of lung cancer effectively. Furthermore, the parameter analysis indicates that it would be highly effective to control the risk of lung cancer by inhibiting the transformation rate from the normal cells to mutated cells and the clonal expansion of cells with fewer gene mutations.

Inference of Molecular Regulatory Systems Using Statistical Path-Consistency Algorithm.

One of the key challenges in systems biology and molecular sciences is how to infer regulatory relationships between genes and proteins using high-throughout omics datasets. Although a wide range of methods have been designed to reverse engineer the regulatory networks, recent studies show that the inferred network may depend on the variable order in the dataset. In this work, we develop a new algorithm, called the statistical path-consistency algorithm (SPCA), to solve the problem of the dependence of variable order. This method generates a number of different variable orders using random samples, and then infers a network by using the path-consistent algorithm based on each variable order. We propose measures to determine the edge weights using the corresponding edge weights in the inferred networks, and choose the edges with the largest weights as the putative regulations between genes or proteins. The developed method is rigorously assessed by the six benchmark networks in DREAM challenges, the mitogen-activated protein (MAP) kinase pathway, and a cancer-specific gene regulatory network. The inferred networks are compared with those obtained by using two up-to-date inference methods. The accuracy of the inferred networks shows that the developed method is effective for discovering molecular regulatory systems.

Risk of lung cancer due to external environmental factor and epidemiological data analysis.

Lung cancer is a cancer with the fastest growth in the incidence and mortality all over the world, which is an extremely serious threat to human's life and health. Evidences reveal that external environmental factors are the key drivers of lung cancer, such as smoking, radiation exposure and so on. Therefore, it is urgent to explain the mechanism of lung cancer risk due to external environmental factors experimentally and theoretically. However, it is still an open issue regarding how external environment factors affect lung cancer risk. In this paper, we summarize the main mathematical models involved the gene mutations for cancers, and review the application of the models to analyze the mechanism of lung cancer and the risk of lung cancer due to external environmental exposure. In addition, we apply the model described and the epidemiological data to analyze the influence of external environmental factors on lung cancer risk. The result indicates that radiation can cause significantly an increase in the mutation rate of cells, in particular the mutation in stability gene that leads to genomic instability. These studies not only can offer insights into the relationship between external environmental factors and human lung cancer risk, but also can provide theoretical guidance for the prevention and control of lung cancer.

A non-linear reverse-engineering method for inferring genetic regulatory networks.

Hematopoiesis is a highly complex developmental process that produces various types of blood cells. This process is regulated by different genetic networks that control the proliferation, differentiation, and maturation of hematopoietic stem cells (HSCs). Although substantial progress has been made for understanding hematopoiesis, the detailed regulatory mechanisms for the fate determination of HSCs are still unraveled. In this study, we propose a novel approach to infer the detailed regulatory mechanisms. This work is designed to develop a mathematical framework that is able to realize nonlinear gene expression dynamics accurately. In particular, we intended to investigate the effect of possible protein heterodimers and/or synergistic effect in genetic regulation. This approach includes the Extended Forward Search Algorithm to infer network structure (top-down approach) and a non-linear mathematical model to infer dynamical property (bottom-up approach). Based on the published experimental data, we study two regulatory networks of 11 genes for regulating the erythrocyte differentiation pathway and the neutrophil differentiation pathway. The proposed algorithm is first applied to predict the network topologies among 11 genes and 55 non-linear terms which may be for heterodimers and/or synergistic effect. Then, the unknown model parameters are estimated by fitting simulations to the expression data of two different differentiation pathways. In addition, the edge deletion test is conducted to remove possible insignificant regulations from the inferred networks. Furthermore, the robustness property of the mathematical model is employed as an additional criterion to choose better network reconstruction results. Our simulation results successfully realized experimental data for two different differentiation pathways, which suggests that the proposed approach is an effective method to infer the topological structure and dynamic property of genetic regulations.

A mathematical model of cell-mediated immune response to tumor.

Mathematical models of tumor-immune interactions provide an analytic framework for studying tumor-immune dynamics. In this paper, we present a mathematical model to describe tumor-immune cell interactions, focusing on the role of the natural killer (NK) cells and CD8+ cytotoxic T lymphocytes (CTLs) in immune surveillance. According to the experimental and clinical results, we determine part of the model parameters to reduce the model parameter space. Then we analyze the local geometric properties of the equilibria of model and carry out numerical simulations to verify the conditions for the stability properties of equilibrium points. Numerical results suggest that the host immune system alone is not fully effective against progression of tumor cells, and CTLs play a crucial role in immune surveillance.

Mathematical modelling the pathway of genomic instability in lung cancer.

Genomic instability plays a significant role in lung cancer. Although substantial research has been conducted using both clinical and theoretical studies, it is still a hotly debated issue to whether genomic instability is necessary or whether genomic instability precedes oncogenes activation and tumor suppressor genes inactivation for lung cancer. In response to this issue, we come up with a mathematical model incorporating effects of genomic instability to investigate the genomic instability pathway of human lung cancer. The presented model are applied to match the incidence rate data of lung cancer from the Life Span Study cohort of the atomic bomb survivors in Nagasaki and Hiroshima and the Surveillance Epidemiology and End Results registry in the United States. Model results suggest that genomic instability is necessary in the tumorigenesis of lung cancer, and genomic instability has no significant impact on the net proliferation rate of cells by statistical criteria. By comparing the results of the LSS data to those of the SEER data, we conclude that the genomic instability pathway exhibits a sensitivity to radiation exposure, more intensive in male patients.

Stochastic modelling of multistage carcinogenesis and progression of human lung cancer.

In this paper, we propose a stochastic multistage model that incorporates clonal expansion of premalignant cells and mutational events. Using the age-specific lung cancer as the test system, the proposed model is used to fit the incidence data in the Surveillance, Epidemiology, and End Results (SEER) registry. We first use the model with different numbers of mutations to fit the data of all lung cancer patients. Our results demonstrate that, although from two to six driver mutations in the genome of lung stem cells are reasonable for normal lung stem cells to become a malignant cell, three driver mutations are most likely to occur in the development of lung cancer. In addition, the models are employed to fit the data of female and male patients separately. The interesting result is that, for female patient data the best fit model contains four mutations while that for male patient data is the three-stage model. Finally, robustness analysis suggests that the decrease of cell net proliferation rates is more effective than the decrease of mutation rates in reducing the lung cancer risk.

The Cost-Effectiveness Analysis and Optimal Strategy of the Tobacco Control.

This paper aims at investigating how the media coverage and smoking cessation treatment should be implemented, for a certain period, to reduce the numbers of smokers and patients caused by smoking while minimizing the total cost. To this end, we first propose a new mathematical model without any control strategies to investigate the dynamic behaviors of smoking. Furthermore, we calculate the basic reproduction number ℛ 0 and discuss the global asymptotic stabilities of the equilibria. Then, from the estimated parameter values, we know that the basic reproduction number ℛ 0 is more than 1, which reveals that smoking is one of the enduring problems of the society. Hence, we introduce two control measures (media coverage and smoking cessation treatment) into the model. Finally, in order to investigate their effects in smoking control and provide an analytical method for the strategic decision-makers, we apply a concrete example to calculate the incremental cost-effectiveness ratios and analyze the cost-effectiveness of all possible combinations of the two control measures. The results indicate that the combination of media coverage and smoking cessation treatment is the most cost-effective strategy for tobacco control.

Instantaneous mutation rate in cancer initiation and progression.

BACKGROUND: Cancer is one of the leading causes for the morbidity and mortality worldwide. Although substantial studies have been conducted theoretically and experimentally in recent years, it is still a challenge to explore the mechanisms of cancer initiation and progression. The investigation for these problems is very important for the diagnosis of cancer diseases and development of treatment schemes. RESULTS: To accurately describe the process of cancer initiation, we propose a new concept of gene initial mutation rate based on our recently designed mathematical model using the non-constant mutation rate. Unlike the widely-used average gene mutation rate that depends on the number of mutations, the gene initial mutation rate can be used to describe the initiation process of a single patient. In addition, we propose the instantaneous tumour doubling time that is a continuous function of time based on the non-constant mutation rate. Our proposed concepts are supported by the clinic data of seven patients with advanced pancreatic cancer. The regression results suggest that, compared with the average mutation rate, the estimated initial mutation rate has a larger value of correlation coefficient with the patient survival time. We also provide the estimated tumour size of these seven patients over time. CONCLUSIONS: The proposed concepts can be used to describe the cancer initiation and progression for different patients more accurately. Since a quantitative understanding of cancer progression is important for clinical treatment, our proposed model and calculated results may provide insights into the development of treatment schemes and also have other clinic implications.

Mutation Mechanisms of Human Breast Cancer.

Cancer is a class of diseases caused by the accumulation of gene mutations. All mutated genes constitute a genetic network for cancer progression. It is very helpful for tumor diagnosis and therapy if we know how many mutated genes are needed for human breast cancer. In this article, we investigate the mutation mechanisms of human breast cancer by modeling the data of surveillance, epidemiology, and end results registry. The data are age-specific incidence rates of breast cancer of females in the United States. We set up stochastic multistage models to estimate the age-specific incidence rates by using several coupled ordinary differential equations derived from the Kolmogorov backward equations. Our results suggest that 2-14 mutations in the genome of breast stem cells are required for normal breast stem cells to become a malignant cell, and 3 gene mutations are most likely to occur in the development of female breast cancer.

The impact of radiation on the development of lung cancer.

Environment factors such as radiation play an important role in the incidence of lung cancer. In spite of substantial efforts in experimental study and mathematical modeling, it is still a significant challenge to estimate lung cancer risk from radiation. To address this issue, we propose a stochastic model to investigate the impact of radiation on the development of lung cancer. The proposed three-stage model with clonal expansion is used to match the data of the male and female patients in the Osaka Cancer Registry (OCR) and Life Span Study (LSS) cohort of atomic bomb survivors in Hiroshima and Nagasaki. Our results indicate that the major effect of radiation on the development of lung cancer is to induce gene mutations for both male and female patients. In particular, for male patients, radiation affects the mutation in normal cells and the transformation from premalignant cells to malignant ones. However, radiation for female patients increases the mutation rates of the first two mutations in the stochastic model. The established relationship between parameters and radiation will provide insightful prediction for the lung cancer incidence in the radiation exposure.

An integrated approach to infer dynamic protein-gene interactions - A case study of the human P53 protein.

Investigating the dynamics of genetic regulatory networks through high throughput experimental data, such as microarray gene expression profiles, is a very important but challenging task. One of the major hindrances in building detailed mathematical models for genetic regulation is the large number of unknown model parameters. To tackle this challenge, a new integrated method is proposed by combining a top-down approach and a bottom-up approach. First, the top-down approach uses probabilistic graphical models to predict the network structure of DNA repair pathway that is regulated by the p53 protein. Two networks are predicted, namely a network of eight genes with eight inferred interactions and an extended network of 21 genes with 17 interactions. Then, the bottom-up approach using differential equation models is developed to study the detailed genetic regulations based on either a fully connected regulatory network or a gene network obtained by the top-down approach. Model simulation error, parameter identifiability and robustness property are used as criteria to select the optimal network. Simulation results together with permutation tests of input gene network structures indicate that the prediction accuracy and robustness property of the two predicted networks using the top-down approach are better than those of the corresponding fully connected networks. In particular, the proposed approach reduces computational cost significantly for inferring model parameters. Overall, the new integrated method is a promising approach for investigating the dynamics of genetic regulation.

A new model of time scheme for progression of colorectal cancer.

BACKGROUND: tumourigenesis can be regarded as an evolutionary process, in which the transformation of a normal cell into a tumour cell involves a number of limiting genetic and epigenetic events. To study the progression process, time schemes have been proposed for studying the process of colorectal cancer based on extensive clinical investigations. Moreover, a number of mathematical models have been designed to describe this evolutionary process. These models assumed that the mutation rate of genes is constant during different stages. However, it has been pointed that the subsequent driver mutations appear faster than the previous ones and the cumulative time to have more driver mutations grows with the growing number of gene mutations. Thus it is still a challenge to calculate the time when the first mutation occurs and to determine the influence of tumour size on the mutation rate. RESULTS: In this work we present a general framework to remedy the shortcoming of existing models. Rather than considering the information of gene mutations based on a population of patients, we for the first time determine the values of the selective advantage of cancer cells and initial mutation rate for individual patients. The averaged values of doubling time and selective advantage coefficient determined by our model are consistent with the predictions made by the published models. Our calculation showed that the values of biological parameters, such as the selective advantage coefficient, initial mutation rate and cell doubling time diversely depend on individuals. Our model has successfully predicted the values of several important parameters in cancer progression, such as the selective advantage coefficient, initial mutation rate and cell doubling time. In addition, experimental data validated our predicted initial mutation rate and cell doubling time. CONCLUSIONS: The introduced new parameter makes our proposed model more flexible to fix various types of information based on different patients in cancer progression.

Mathematical modeling of GATA-switching for regulating the differentiation of hematopoietic stem cell.

BACKGROUND: Hematopoiesis is a highly orchestrated developmental process that comprises various developmental stages of the hematopoietic stem cells (HSCs). During development, the decision to leave the self-renewing state and selection of a differentiation pathway is regulated by a number of transcription factors. Among them, genes GATA-1 and PU.1 form a core negative feedback module to regulate the genetic switching between the cell fate choices of HSCs. Although extensive experimental studies have revealed the mechanisms to regulate the expression of these two genes, it is still unclear how this simple module regulates the genetic switching. METHODS: In this work we proposed a mathematical model to study the mechanisms of the GATA-PU.1 gene network in the determination of HSC differentiation pathways. We incorporated the mechanisms of GATA switch into the module, and developed a mathematical model that comprises three genes GATA-1, GATA-2 and PU.1. In addition, a novel multiple-objective optimization method was designed to infer unknown parameters in the proposed model by realizing different experimental observations. A stochastic model was also designed to describe the critical function of noise, due to the small copy numbers of molecular species, in determining the differentiation pathways. RESULTS: The proposed deterministic model has successfully realized three stable steady states representing the priming and different progenitor cells as well as genetic switching between the genetic states under various experimental conditions. Using different values of GATA-1 synthesis rate for the GATA-1 protein availability in the chromatin sites during the time period of GATA switch, stochastic simulations for the first time have realized different proportions of cells leading to different developmental pathways under various experimental conditions. CONCLUSIONS: Mathematical models provide testable predictions regarding the mechanisms and conditions for realizing different differentiation pathways of hematopoietic stem cells. This work represents the first attempt at using a discrete stochastic model to realize the decision of HSC differentiation pathways showing a multimodal distribution.

Chemical memory reactions induced bursting dynamics in gene expression.

Memory is a ubiquitous phenomenon in biological systems in which the present system state is not entirely determined by the current conditions but also depends on the time evolutionary path of the system. Specifically, many memorial phenomena are characterized by chemical memory reactions that may fire under particular system conditions. These conditional chemical reactions contradict to the extant stochastic approaches for modeling chemical kinetics and have increasingly posed significant challenges to mathematical modeling and computer simulation. To tackle the challenge, I proposed a novel theory consisting of the memory chemical master equations and memory stochastic simulation algorithm. A stochastic model for single-gene expression was proposed to illustrate the key function of memory reactions in inducing bursting dynamics of gene expression that has been observed in experiments recently. The importance of memory reactions has been further validated by the stochastic model of the p53-MDM2 core module. Simulations showed that memory reactions is a major mechanism for realizing both sustained oscillations of p53 protein numbers in single cells and damped oscillations over a population of cells. These successful applications of the memory modeling framework suggested that this innovative theory is an effective and powerful tool to study memory process and conditional chemical reactions in a wide range of complex biological systems.

Mathematical modelling of stem cell differentiation: the PU.1-GATA-1 interaction.

The transcription factors PU.1 and GATA-1 are known to be important in the development of blood progenitor cells. Specifically they are thought to regulate the differentiation of progenitor cells into the granulocyte/macrophage lineage and the erythrocyte/megakaryocite lineage. While several mathematical models have been proposed to investigate the interaction between the transcription factors in recent years, there is still debate about the nature of the progenitor state in the dynamical system, and whether the existing models adequately capture new knowledge about the interactions gleaned from experimental data. Further, the models utilise different formalisms to represent the genetic regulation, and it appears that the resulting dynamical system depends upon which formalism is adopted. In this paper we analyse the four existing models, and propose an alternative model which is shown to demonstrate a rich variety of dynamical systems behaviours found across the existing models, including both bistability and tristability required for modelling the undifferentiated progenitors.