Nonlinear Ordinary Differential Equation Model Research Articles

Abstract B034: Order-of-mutation effects on cancer progression: Models for myeloproliferative neoplasm

Abstract In some patients with myeloproliferative neoplasms (MPN), two genetic mutations are often found, JAK2 V617F and one in the TET2 gene. Whether or not one mutation is present will influence how the other subsequent mutation affects the regulation of gene expression. When both mutations are present, the order of their occurrence has been shown to influence disease progression and prognosis. We propose a nonlinear ordinary differential equation (ODE) model and a Moran process models to explain the non-additive and non-commutative mutation effects on recent clinical observations of gene expression patterns, proportions of cells with different mutations, and ages at diagnosis of MPN. These observations consistently shape our modeling framework. Our key proposal is that bistability in gene expression provides a natural explanation for many observed order-of-mutation effects. We also propose potential experimental measurements that can be used to confirm or refute predictions of our models. Citation Format: Yue Wang, Tom Chou, Blerta Shtylla. Order-of-mutation effects on cancer progression: Models for myeloproliferative neoplasm [abstract]. In: Proceedings of the AACR Special Conference in Cancer Research: Translating Cancer Evolution and Data Science: The Next Frontier; 2023 Dec 3-6; Boston, Massachusetts. Philadelphia (PA): AACR; Cancer Res 2024;84(3 Suppl_2):Abstract nr B034.

ILSGRN: inference of large-scale gene regulatory networks based on multi-model fusion.

Gene regulatory networks (GRNs) are a way of describing the interaction between genes, which contribute to revealing the different biological mechanisms in the cell. Reconstructing GRNs based on gene expression data has been a central computational problem in systems biology. However, due to the high dimensionality and non-linearity of large-scale GRNs, accurately and efficiently inferring GRNs is still a challenging task. In this article, we propose a new approach, iLSGRN, to reconstruct large-scale GRNs from steady-state and time-series gene expression data based on non-linear ordinary differential equations. Firstly, the regulatory gene recognition algorithm calculates the Maximal Information Coefficient between genes and excludes redundant regulatory relationships to achieve dimensionality reduction. Then, the feature fusion algorithm constructs a model leveraging the feature importance derived from XGBoost (eXtreme Gradient Boosting) and RF (Random Forest) models, which can effectively train the non-linear ordinary differential equations model of GRNs and improve the accuracy and stability of the inference algorithm. The extensive experiments on different scale datasets show that our method makes sensible improvement compared with the state-of-the-art methods. Furthermore, we perform cross-validation experiments on the real gene datasets to validate the robustness and effectiveness of the proposed method. The proposed method is written in the Python language, and is available at: https://github.com/lab319/iLSGRN.

Likelihood-ratio test statistic for the finite-sample case in nonlinear ordinary differential equation models.

Likelihood ratios are frequently utilized as basis for statistical tests, for model selection criteria and for assessing parameter and prediction uncertainties, e.g. using the profile likelihood. However, translating these likelihood ratios into p-values or confidence intervals requires the exact form of the test statistic's distribution. The lack of knowledge about this distribution for nonlinear ordinary differential equation (ODE) models requires an approximation which assumes the so-called asymptotic setting, i.e. a sufficiently large amount of data. Since the amount of data from quantitative molecular biology is typically limited in applications, this finite-sample case regularly occurs for mechanistic models of dynamical systems, e.g. biochemical reaction networks or infectious disease models. Thus, it is unclear whether the standard approach of using statistical thresholds derived for the asymptotic large-sample setting in realistic applications results in valid conclusions. In this study, empirical likelihood ratios for parameters from 19 published nonlinear ODE benchmark models are investigated using a resampling approach for the original data designs. Their distributions are compared to the asymptotic approximation and statistical thresholds are checked for conservativeness. It turns out, that corrections of the likelihood ratios in such finite-sample applications are required in order to avoid anti-conservative results.

Real-time adaptive sparse-identification-based predictive control of nonlinear processes

This study introduces a sparse identification-based model predictive control (MPC) framework that incorporates on-line updates of the sparse-identified model to account for nonlinear dynamics and model uncertainty in process systems. The methodology involves obtaining a nonlinear first-order ordinary differential equation model using sparse identification for nonlinear dynamics (SINDy), which is integrated into two control schemes: Lyapunov-based MPC (LMPC) for achieving steady-state operation and Lyapunov-based economic MPC (LEMPC) for achieving both closed-loop stability and optimal economic performance. To improve prediction accuracy, an on-line model update scheme is proposed for the SINDy models. Specifically, an error-trigger mechanism that utilizes prediction errors and then uses the most recent process data to update the parameters of the SINDy model in real-time is designed. By incorporating the error-triggered on-line model updates in the SINDy-based LMPC and LEMPC, the dynamic performance of the process is enhanced, ensuring closed-loop stability, optimality, and smooth control actions. Following theoretical results on the boundedness of the closed-loop states and detailed discussions on the selection criteria for parameters of the error-triggered SINDy update scheme, the effectiveness of the proposed methodology is demonstrated through a chemical process example with time-varying disturbances under the LEMPC framework.

MATHEMATICAL MODELING OF STEM CELL THERAPY FOR LEFT VENTRICULAR REMODELING AFTER MYOCARDIAL INFARCTION

The heart is an organ with a limited capacity for regeneration and repair. In this paper, a new mathematical model is presented to study the left ventricular remodeling after myocardial infarction (MI) and followed stem cell therapeutic effort. The model represents the post-MI regeneration process of cardiomyocytes under stem cell therapy with oxygen restoration. The resulting system of nonlinear ordinary differential equations (ODE) is studied numerically in order to demonstrate the functionality and performance of the new model. The optimal time of stem cell injection for various oxygen restorations is determined. Moreover, the regeneration of cardiomyocytes is successfully correlated with improved left ventricle function observed in experiments. The proposed nonlinear ODE model is able to capture the complicated biological interactions in post-MI remodeling and can serve as a platform for in silico simulation and perturbation to optimize MI stem cell therapy.

Likelihood-ratio test statistic for the finite-sample case in nonlinear ordinary differential equation models

Likelihood-ratio test statistic for the finite-sample case in nonlinear ordinary differential equation models

On the Optimal Control of HIV-TB Co-Infection and Improvement of Workplace Productivity

Human immunodeficiency virus (HIV) and tuberculosis (TB) have long been known to have a synergistic relationship. This is a result of each of the diseases impacting negatively on the immune system of the infected persons. The impact of these diseases on workforce productivity is studied in this paper from the viewpoint of dynamical systems. In this paper, we present a nonlinear ordinary differential equation model to study the dynamics of HIV-TB co-infection and its effect on workforce productivity. The main model is first decoupled into two basic submodels of HIV-only and TB-only models, whose qualitative properties are presented before the qualitative properties of the main model are studied. While the HIV-only model is shown to have a globally asymptotically stable disease-free equilibrium whenever its basic reproduction number is less than unity, the TB-only model is shown to exhibit backward bifurcation under some conditions. To investigate the impact of various intervention strategies on the control of the co-infection and improvement of workforce productivity, five time-dependent controls (involving transmission prevention for the two diseases, therapy for the two diseases, and capacity building for improved workforce productivity) are incorporated into the basic model to form an optimal control problem, which is qualitatively analyzed using Pontryagin’s maximum principle and numerically simulated. Incremental cost-effectiveness analysis is conducted with the results of the numerical simulations. It is observed that the most cost-effective strategy for fighting the spread of the co-infection with enhanced productivity is that of combining both preventative and curative measures along with skills training.

Dissipativity of nonlinear ODE model of distribution voltage profile

In this paper, we consider a power distribution system consisting of a straight feeder line. A nonlinear ordinary differential equation (ODE) model is used to describe the voltage distribution profile over the feeder line. At first, we show the dissipativity of the subsystems corresponding to active and reactive powers. We also show that the dissipation rates of these subsystems coincide with the distribution loss given by a square of current amplitudes. Moreover, the entire distribution system is decomposed into two subsystems corresponding to voltage amplitude and phase. As a main result, we prove the dissipativity of these subsystems based on the decomposition. As a physical interpretation of these results, we clarify that the phenomena related to the gradients of the voltage amplitude and phase are induced in a typical power distribution system from the dissipation equalities. Finally, we discuss a reduction of distribution losses based on the dissipation rate of the subsystem of voltage amplitude.

Asymptotic assessment of distribution voltage profile using a nonlinear ODE model

The promising increase of Electric Vehicles (EVs) in our society poses a challenging problem on the impact assessment of their charging/discharging to power distribution grids. This paper addresses the assessment problem in a framework of nonlinear differential equations. Specifically, we address the nonlinear ODE (Ordinary Differential Equation) model for representing the spatial profile of voltage phasor along a distribution feeder, which has been recently introduced in literature. The assessment problem is then formulated as a two-point boundary value problem of the nonlinear ODE model. In this paper we derive an asymptotic charcterisation of solutions of the problem through the standard regular perturbation method. This provides a mathematically-rigor and quantitative method for assessing how the charging/discharging of EVs affects the spatial profile of distribution voltage. Effectiveness of the asymptotic charcterisation is established with simulations of both simple and practical configurations of the power distribution grid.

Mathematical Treatment Model of Lassa Fever

We present and explore a novel mathematical treatment model of the epidemiology of Lassa Fever (LF). The model is a system of nonlinear ordinary differential equation model for rodent and human population. We analyzed the model to find the stability of the disease-free equilibrium and test which model parameters affect this stability most significantly. The purpose of this paper is to investigate the impact of treatment on the control of LF. The analysis revealed that treatment rate of humans will have a positive impact in reducing the burden of LF in the population. Our model predicts that treatment control can reduce the population level transmission by up to 12% alone without existing interventions. Therefore, treatment has significant effect on LF transmission, but it may not be able to eliminate the disease unless a multiple control strategy is adopted. Finally, some numerical simulations were carried out to support our theoretical results.

A nonlinear sparse neural ordinary differential equation model for multiple functional processes.

In this article, we propose a new sparse neural ordinary differential equation (ODE) model to characterize flexible relations among multiple functional processes. We characterize the latent states of the functions via a set of ordinary differential equations. We then model the dynamic changes of the latent states using a deep neural network (DNN) with a specially designed architecture and a sparsity-inducing regularization. The new model is able to capture both nonlinear and sparse dependent relations among multivariate functions. We develop an efficient optimization algorithm to estimate the unknown weights for the DNN under the sparsity constraint. We establish both the algorithmic convergence and selection consistency, which constitute the theoretical guarantees of the proposed method. We illustrate the efficacy of the method through simulations and a gene regulatory network example.

Machine learning methods for probabilistic locked-mode predictors in tokamak plasmas

A rotating tokamak plasma can interact resonantly with the external helical magnetic perturbations, also known as error fields. This can lead to locking and then to disruptions. We leverage machine learning (ML) methods to predict the locking events. We use a coupled third-order nonlinear ordinary differential equation model to represent the interaction of the magnetic perturbation and the plasma rotation with the error field. This model is sufficient to describe qualitatively the locking and unlocking bifurcations. We explore using ML algorithms with the simulation data and experimental data, focusing on the methods that can be used with sparse datasets. These methods lead to the possibility of the avoidance of locking in real-time operations. We describe the operational space in terms of two control parameters: the magnitude of the error field and the rotation frequency associated with the momentum source that maintains the plasma rotation. The outcomes are quantified by order parameters that completely characterize the state, whether locked or unlocked. We use unsupervised ML methods to classify locked/unlocked states and note the usefulness of a certain normalization of the order parameters. Three supervised ML classifiers are used in suite to estimate the probability of locking in the region of control parameter space with hysteresis, i.e., the set of control parameters for which both locked and unlocked states can exist. The results show that a neural network gives the best estimate of the locking probability. An analogy of the present locking model with the van der Waals equation of state is also provided.

Internal energy activation stimulus on magneto-bioconvective Powell-Eyring nanofluid containing gyrotactic microorganisms under active/passive nanoparticles flux

Influence of Stephan blowing on a three-dimensional hydro magneto-bioconvective Eyring-Powell nanofluid containing Gyrotactic microorganisms under active and passive nanoparticle flux conditions is investigated in this paper. The presence of non-linear radiation along a bidirectional stretched surface is also deliberated throughout this analysis. A comparative study is made up for active and passive nanoparticle volume control. The study is relevant to novel microbial fuel cell technologies combining the nanofluid with bioconvection phenomena. The Prandtl’s boundary layer equations, approximated by Oberbeck-Boussinesq’s are studied under suitable boundary conditions. Similarity transformation is used to transform the governing boundary layer equations to dimensionless nonlinear ordinary differential equation model. The non-linear model is then resolved by combining the Runge–Kutta method and the MATHEMATICA software. The dimensionless velocity, temperature, nanoparticle concentration and density of motile microorganisms together with the wall shear stress, Nusselt, Sherwood and density of motile microorganism number are graphically presented to visualize the effects of particular parameters. A novel idea of Stefan blowing on three-dimensional space taking into consideration the non-linear radiation is implemented in the investigation of Eyring-Powell flow over a deformable sheet under the effects of slip and activation energy for the first time. Axial Newtonian slip decreases the momentum of the nanoflow while tangential slip augments the flow. Bio-convection parameter reduces the nanoparticle concentration of the rescaled density of motile microorganisms. Skin friction factor is dominant in passive nanoparticles flux rather than active nanoparticles flux. Passive nanoparticles help in smooth delivery of drugs which is seen here for zero mass flux. Energy transfer rate is high compared to active nanoparticle volume control. A zero mass flux acts as a cooling agent in this study by diffusing more heat from the system.

The effect of time ordering and concurrency in a mathematical model of chemoradiotherapy

We study the effect of switching the order of administration of cytotoxic drugs and radiation in cancer therapy by comparing a sequential and a concurrent protocol of chemoradiation. For this purpose, we derive a nonlinear ordinary differential equation model based on well-accepted knowledge of chemotherapy and radiotherapy for in vitro solid tumors. Using the bifurcation theory, we demonstrate that the administration of radiotherapy concurrently, once the chemotherapeutic regime has caused a considerable reduction of the tumor burden, might be more pertinent than the reverse strategy in those circumstances where bistability occurs. Consequently, our analysis suggests that an adequate time order and concurrency may have potential benefits in chemoradiation of solid tumors.

Modeling and Analysis of Mood Dynamics in the Bipolar Spectrum

This article introduces a nonlinear ordinary differential equation model of mood dynamics for disorders on the bipolar spectrum. Motivated by biopsychosocial findings, the model characterizes mood as a 2-D state corresponding to manic and depressive features, enabling the representation of most diagnoses of bipolar and depressive disorders. We perform a mathematical analysis of conditions for the mood to stabilize to euthymia and discuss its psychotherapeutic implications. Furthermore, a computational analysis applied to pharmacotherapy depicts a mechanism that results in a switch from depression to mania when the bipolar disorder was misdiagnosed as major depressive disorder, and an antidepressant is administered without a mood stabilizer. This work innovates by offering a concise representation of most features of mood disorders in existing mathematical models, providing a framework for studying dynamics in the bipolar spectrum.

Deterministic modeling for HIV and aids epidemics with viral load detectability

We proposed and analyzed a nonlinear ordinary differential equation model for HIV and AIDS epidemics with viral load detectability and derived the interaction mechanisms between model compartments. The model is studied qualitatively and obtain a threshod parameter that represents the largest eigenvalue by using next-generation approach. Furthermore, local and global stability conditions for disease free and endemic equilibria are determined, similarly, we perfomed birfucation analysis and sensitivity analysis of the model. In addition, using the numerical results, we showed the condition on which the disease can die out (R_0^d 1). Lastly, the results showed that people living with HIV to lower the viral loads to Undetectable level by taking the prescribed dozes as shown in Figure (7) resulting to viral suppression.

A comparison of approximate versus exact techniques for Bayesian parameter inference in nonlinear ordinary differential equation models.

The behaviour of many processes in science and engineering can be accurately described by dynamical system models consisting of a set of ordinary differential equations (ODEs). Often these models have several unknown parameters that are difficult to estimate from experimental data, in which case Bayesian inference can be a useful tool. In principle, exact Bayesian inference using Markov chain Monte Carlo (MCMC) techniques is possible; however, in practice, such methods may suffer from slow convergence and poor mixing. To address this problem, several approaches based on approximate Bayesian computation (ABC) have been introduced, including Markov chain Monte Carlo ABC (MCMC ABC) and sequential Monte Carlo ABC (SMC ABC). While the system of ODEs describes the underlying process that generates the data, the observed measurements invariably include errors. In this paper, we argue that several popular ABC approaches fail to adequately model these errors because the acceptance probability depends on the choice of the discrepancy function and the tolerance without any consideration of the error term. We observe that the so-called posterior distributions derived from such methods do not accurately reflect the epistemic uncertainties in parameter values. Moreover, we demonstrate that these methods provide minimal computational advantages over exact Bayesian methods when applied to two ODE epidemiological models with simulated data and one with real data concerning malaria transmission in Afghanistan.

Overview of Gene Regulatory Network Inference Based on Differential Equation Models.

Reconstruction of gene regulatory networks (GRN) plays an important role in understanding the complexity, functionality and pathways of biological systems, which could support the design of new drugs for diseases. Because differential equation models are flexible androbust, these models have been utilized to identify biochemical reactions and gene regulatory networks. This paper investigates the differential equation models for reverse engineering gene regulatory networks. We introduce three kinds of differential equation models, including ordinary differential equation (ODE), time-delayed differential equation (TDDE) and stochastic differential equation (SDE). ODE models include linear ODE, nonlinear ODE and S-system model. We also discuss the evolutionary algorithms, which are utilized to search the optimal structures and parameters of differential equation models. This investigation could provide a comprehensive understanding of differential equation models, and lead to the discovery of novel differential equation models.

Robust Calibration of High Dimension Nonlinear Dynamical Models for Omics Data: An Application in Cancer Systems Biology

In computational mathematical modeling of biological systems, most model parameters, such as initial conditions, kinetics, and scale factors, are usually unknown because they cannot be directly measured. Therefore, key issues in system identification of nonlinear systems are model calibration and identifiability analysis. Currently, existing methodologies for parameter estimation are divided in two classes: frequentist and Bayesian methods. The first optimize a cost function, while the second estimate the posterior distribution of parameters through different sampling techniques. However, when dealing with high-dimensional models, these methodologies suffer from an increasing computational cost due to the important volume of -omic data necessary to carry out reliable and robust solutions. Here, we present an innovative Bayesian method, called conditional robust calibration (CRC), for model calibration and identifiability analysis. The algorithm is an iterative procedure based on a uniform and joint perturbation of the parameter space. At each step the algorithm returns the probability density functions of all parameters that progressively shrink toward specific points in the parameter space. These distributions are estimated on parameter samples that guarantee a certain level of agreement between each observable and the corresponding in silico measure. We apply CRC to a nonlinear high-dimensional ordinary differential equations model representing the signaling pathway of p38MAPK in multiple myeloma. The available data set consists of time courses of proteomic cancerous data. We test CRC performances in comparison with profile-likelihood and approximate Bayesian computation sequential Monte Carlo. We obtain a more precise and robust solution with a reduced computational cost.

CRA toolbox: software package for conditional robustness analysis of cancer systems biology models in MATLAB

BackgroundIn cancer research, robustness of a complex biochemical network is one of the most relevant properties to investigate for the development of novel targeted therapies. In cancer systems biology, biological networks are typically modeled through Ordinary Differential Equation (ODE) models. Hence, robustness analysis consists in quantifying how much the temporal behavior of a specific node is influenced by the perturbation of model parameters. The Conditional Robustness Algorithm (CRA) is a valuable methodology to perform robustness analysis on a selected output variable, representative of the proliferation activity of cancer disease.ResultsHere we introduce our new freely downloadable software, the CRA Toolbox. The CRA Toolbox is an Object-Oriented MATLAB package which implements the features of CRA for ODE models. It offers the users the ability to import a mathematical model in Systems Biology Markup Language (SBML), to perturb the model parameter space and to choose the reference node for the robustness analysis. The CRA Toolbox allows the users to visualize and save all the generated results through a user-friendly Graphical User Interface (GUI). The CRA Toolbox has a modular and flexible architecture since it is designed according to some engineering design patterns. This tool has been successfully applied in three nonlinear ODE models: the Prostate-specific Pten−/− mouse model, the Pulse Generator Network and the EGFR-IGF1R pathway.ConclusionsThe CRA Toolbox for MATLAB is an open-source tool implementing the CRA to perform conditional robustness analysis. With its unique set of functions, the CRA Toolbox is a remarkable software for the topological study of biological networks. The source and example code and the corresponding documentation are freely available at the web site: http://gitlab.ict4life.com/SysBiOThe/CRA-Matlab.