Tianhai Tian’s research while affiliated with Monash University (Australia) and other places

Angiogenesis plays an important role in the development of tumor since it allows for the delivery of oxygen and nutrients. Understanding the dynamics of angiogenesis is essential for accurately characterizing tumor growth during the vascular phase and for developing models that optimize therapeutic strategies. In order to provide beneficial information and theoretical models for clinical research of tumor therapy strategies, a new mathematical model is established to describe the interaction between vascular epithelial cells and tumor cells. Through the stability analysis of the equilibria, the results show that in the absence of therapeutic intervention, the number of tumor cells will naturally stabilize at a positive value. Further, the anti-angiogenesis treatment term is added to the model to investigate the effects of anti-vascular therapy. The results indicate that anti-angiogenic therapy alone cannot completely eliminate a tumor, but can observably reduce the number of the tumor. Finally, numerical simulations are carried out to investigate the therapeutic effect of anti-vascular therapy combined with traditional treatment. The numerical simulation results show that the anti-angiogenesis therapy is a good adjunct to traditional therapy, which is beneficial to the design of better tumor treatment program.

Manifold learning techniques have emerged as crucial tools for uncovering latent patterns in high-dimensional single-cell data. However, most existing dimensionality reduction methods primarily rely on 2D visualization, which can distort true data relationships and fail to extract reliable biological information. Here, we present DTNE (diffusive topology neighbor embedding), a dimensionality reduction framework that faithfully approximates manifold distance to enhance cellular relationships and dynamics. DTNE constructs a manifold distance matrix using a modified personalized PageRank algorithm, thereby preserving topological structure while enabling diverse single-cell analyses. This approach facilitates distribution-based cellular relationship analysis, pseudotime inference, and clustering within a unified framework. Extensive benchmarking against mainstream algorithms on diverse datasets demonstrates DTNE’s superior performance in maintaining geodesic distances and revealing significant biological patterns. Our results establish DTNE as a powerful tool for high-dimensional data analysis in uncovering meaningful biological insights.

The mitogen-activated protein kinase (MAPK) pathway is an important intracellular signaling cascade that plays a key role in various cellular processes. Understanding the regulatory mechanisms of this pathway is essential for developing effective interventions and targeted therapies for related diseases. Recent advances in single-cell proteomic technologies have provided unprecedented opportunities to investigate the heterogeneity and noise within complex, multi-signaling networks across diverse cells and cell types. Mathematical modeling has become a powerful interdisciplinary tool that bridges mathematics and experimental biology, providing valuable insights into these intricate cellular processes. In addition, statistical methods have been developed to infer pathway topologies and estimate unknown parameters within dynamic models. This review presents a comprehensive analysis of how mathematical modeling of the MAPK pathway deepens our understanding of its regulatory mechanisms, enhances the prediction of system behavior, and informs experimental research, with a particular focus on recent advances in modeling and inference using single-cell proteomic data.

Drug resistance is one of the most intractable issues in targeted therapy for cancer diseases. It has also been demonstrated to be related to cancer heterogeneity, which promotes the emergence of treatment-refractory cancer cell populations. Focusing on how cancer cells develop resistance during the encounter with targeted drugs and the immune system, we propose a mathematical model for studying the dynamics of drug resistance in a conjoint heterogeneous tumor-immune setting. We analyze the local geometric properties of the equilibria of the model. Numerical simulations show that the selectively targeted removal of sensitive cancer cells may cause the initially heterogeneous population to become a more resistant population. Moreover, the decline of immune recruitment is a stronger determinant of cancer escape from immune surveillance or targeted therapy than the decay in immune predation strength. Sensitivity analysis of model parameters provides insight into the roles of the immune system combined with targeted therapy in determining treatment outcomes.

Complex molecular details of transcriptional regulation can be coarse-grained by assuming that reaction waiting times for promoter-state transitions, the mRNA synthesis, and the mRNA degradation follow general distributions. However, how such a generalized two-state model is analytically solved is a long-standing issue. Here we first present analytical formulas of burst-size distributions for this model. Then, we derive an iterative equation for the mRNA moment-generating function, by which mRNA raw and binomial moments of any order can be conveniently calculated. The analytical results obtained in the special cases of phase-type waiting-time distributions not only provide insights into the mechanisms of complex transcriptional regulations but also bring conveniences for experimental data-based statistical inferences.

Stochastic differential equations (SDEs) are a powerful tool to model fluctuations and uncertainty in complex systems. Although numerical methods have been designed to simulate SDEs effectively, it is still problematic when numerical solutions may be negative, but application problems require positive simulations. To address this issue, we propose balanced implicit Patankar–Euler methods to ensure positive simulations of SDEs. Instead of considering the addition of balanced terms to explicit methods in existing balanced methods, we attempt the deletion of possible negative terms from the explicit methods to maintain positivity of numerical simulations. The designed balanced terms include negative-valued drift terms and potential negative diffusion terms. The proposed method successfully addresses the issue of divisions with very small denominators in our recently designed stochastic Patankar method. Stability analysis shows that the balanced implicit Patankar–Euler method has much better stability properties than our recently designed composite Patankar–Euler method. Four SDE systems are used to examine the effectiveness, accuracy, and convergence properties of balanced implicit Patankar–Euler methods. Numerical results suggest that the proposed balanced implicit Patankar–Euler method is an effective and efficient approach to ensure positive simulations when any appropriate stepsize is used in simulating SDEs of biological regulatory systems.

Drug resistance is one of the most intractable issues associated with cancer treatment in clinical practice. Mathematical models provide an analytic framework for facilitating the understanding of resistance evolution dynamics and the design of cancer clinical trial. In this paper, we develop an elementary, compartmental mathematical model for absolute drug resistance, focusing on the effects of point mutations in genetic drivers of malignancy. A set of ordinary differential equations (ODEs) is used to describe the dynamics of competing heterogeneous cancer cell populations while taking account of pharmacokinetics. All possible equilibria and their local geometric properties are analyzed, with the result suggests that the system exhibits bistable dynamics. The existence of optimal treatment time is discussed. To identify the critical parameters which influence cellular dynamics, we also perform parameter sensitivity analysis. Finally, numerical simulations are presented to verify the feasibilities of our analytical results and to find that the pre-existence of resistant cell phenotypes contributes more than resistant mutants generated during the treatment phase.

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.