Linear and nonlinear mathematical models for noninvasive ventilation

Two mathematical models are proposed for passive, noninvasive ventilation. Both models take the form of coupled ordinary differential equations that describe the volume in a single compartment lung. One model is linear and the other nonlinear; both models are derived from basic pressure balances in the lung-ventilator system. These models are also compared to a physical model using a test lung. Both the physical and mathematical models exhibit instabilities that appear to have important clinical implications. The simulations from these models, and the forms of their governing equations, suggest that the presence of an airway leak proximal to the airway opening during pressure support noninvasive ventilation may render this mode of ventilation dynamically unstable. The mathematical models are extended to incorporate a special type of nonpassive ventilation where the total cycle times of the ventilator depend on the inspiratory phases of these cycles.     

Experimental versus numerical data for breast cancer progression

This paper deals with a mouse model of breast cancer based on two mammary adenocarcinoma cell lines derived from a spontaneous tumor of the mammary gland in a female BALB/c mouse. We investigate both animal and mathematical models of tumor progression, and demonstrate a correspondence between the experimental and predicted data. The mathematical model is solved numerically and the laboratory data are utilized in order to find unknown parameters for the model equations. The results of the numerical experiments illustrate that the mathematical model has a potential to describe the growth of cancer cells in vivo.     

Nonlinear modeling with mammographic evidence of carcinoma

The goal of this paper is to apply oncological data for mathematical modeling of breast cancer progression. The studied model is composed of nonlinear partial integro-differential equations, which are formulated with unknown parameters. It is demonstrated that it is possible to find such parameter values for the nonlinear model so that its solutions correspond to the oncological data, therefore showing the potential and extending the applications of the model to breast cancer. The nonlinear model equations are solved numerically and the numerical results confirm the oncological data.     

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??The multivariate response is commonly seen in longitudinal and cross-sectional design. The marginal model is an important tool in discovering the average influence of the covariates on the response. A main feature of the marginal model is that even without specifying the inter-correlation among different components of the response, we still get consistent estimation of the regression parameters. This paper discusses the GMM estimation of marginal model when the covariates are missing at random. Using the inverse probability weighting and different basic working correlation matrices, we obtain a series of estimating equations. We estimate the parameters of interest by minimizing the corresponding quadratic inference function. Asymptotic normality of the proposed estimator is established. Simulation studies are conducted to investigate the finite sample performance of the new estimator. We also apply our proposal to a real data of mathematical achievement from middle school students.     

Analysis of tumor as an inverse problem provides a novel theoretical framework for understanding tumor biology and therapy

We use a novel “inverse problem” technique to construct a basic mathematical model of the interacting populations at the tumor-host interface. This approach assumes that invasive cancer is a solution to the set of state equations that govern the interactions of transformed and normal cells. By considering the invading tumor edge as a traveling wave, the general form of the state equations can be inferred. The stability of this traveling wave solution imposes constraints on key biological quantities which appear as parameters in the model equations. Based on these constraints, we demonstrate the limitations of traditional therapeutic strategies in clinical oncology that focus solely on killing tumor cells or reducing their rate of proliferation. The results provide insights into fundamental mechanisms that may prevent these approaches from successfully eradicating most common cancers despite several decades of research. Alternative therapies directed at modifying the key parameters in the state equations to destabilize the propagating solution are proposed.     

An approach to benchmarking of loosely coupled low-cost navigation systems

New solutions to the navigation problem related to low-cost integrated navigation systems (INS) are often published. Since these new solutions are generally compared with ad hoc mathematical models that are not fully exposed, one cannot be sure of the relative improvements. In this work, complete mathematical model for a low-cost INS is suggested to be used as a benchmarking. As far as the authors’ knowledge, a benchmarking for low-cost INS has not been previously reported. Shown INS comprises a strapdown inertial navigation system, loosely coupled to a GPS receiver. The INS mathematical model is based upon classical navigation equations and classical sensor models, both from recognized authors. The algorithm that details the INS operation is also presented. The benchmarking is provided as an open-source toolbox for MATLAB. Additionally, this work can be taken as a starting point for new practitioners in the INS field. To validate the INS mathematical model, real-world data sets from three different Micro Electro-Mechanical Systems (MEMS) inertial measurement units (IMU) and a GPS receiver are processed. It is observed that obtained RMS errors from the three INS are coherent with the quality of corresponding MEMS IMU. This confirms that the proposed benchmarking is a suitable tool to evaluate objectively new solutions to low-cost INS.     

A 8-neighbor model lattice Boltzmann method applied to mathematical–physical equations

A lattice Boltzmann method (LBM) 8-neighbor model (9-bit model) is presented to solve mathematical–physical equations, such as, Laplace equation, Poisson equation, Wave equation and Burgers equation. The 9-bit model has been verified by several test cases. Numerical simulations, including 1D and 2D cases, of each problem are shown, respectively. Comparisons are made between numerical predictions and analytic solutions or available numerical results from previous researchers. It turned out that the 9-bit model is computationally effective and accurate for all different mathematical–physical equations studied. The main benefits of the new model proposed is that it is faster than the previous existing models and has a better accuracy.     

The odd log-logistic Topp–Leone G family of distributions: heteroscedastic regression models and applications

We introduce a new class of distributions and provide a comprehensive treatment of its mathematical properties. The maximum likelihood method is discussed to estimate the parameters of the new model by means of Monte-Carlo simulation study. The heteroscedastic regression models with long-term survival are introduced to model data sets with the non homogeneity of the error variances in the presence of cured individuals. The potentiality of the proposed models is illustrated by means of four real data sets.     

电力系统中地网腐蚀诊断的一种新的数学模型及其仿真计算

对电力系统中具有重大应用价值的地网腐蚀诊断问题抽象出仿真求解的一种新的数学模型:即求解带约束的非线性隐式方程组模型.但由于问题本身的物理特性决定了所建立的数学模型具有以下特点:一是非线性方程组为欠定方程组,而且非线性程度非常高;二是方程组的所有函数均为隐函数;三是方程组附加若干箱约束条件.这种特性给模型分析与算法设计带来巨大困难.对于欠定方程组的求解,文中根据工程实际背景,尽可能地扩充方程的个数,使之成为超定方程组,然后对欠定方程组和超定方程组分别求解并进行比较.将带约束的非线性隐函数方程组求解问题,转化为无约束非线性最小二乘问题,并采用矩阵求导等技术和各种算法设计技巧克服隐函数的计算困难,最后使用拟牛顿信赖域方法进行计算.大量的计算实例表明,文中所提出的数学模型及求解方法是可行的.与目前广泛采用的工程简化模型相比较,在模型和算法上具有很大优势.     

Learning patient-specific parameters for a diffuse interface glioblastoma model from neuroimaging data

Parameters in mathematical models for glioblastoma multiforme (GBM) tumour growth are highly patient specific. Here, we aim to estimate parameters in a Cahn–Hilliard type diffuse interface model in an optimised way using model order reduction (MOR) based on proper orthogonal decomposition (POD). Based on snapshots derived from finite element simulations for the full-order model (FOM), we use POD for dimension reduction and solve the parameter estimation for the reduced-order model (ROM). Neuroimaging data are used to define the highly inhomogeneous diffusion tensors as well as to define a target functional in a patient-specific manner. The ROM heavily relies on the discrete empirical interpolation method, which has to be appropriately adapted in order to deal with the highly nonlinear and degenerate parabolic partial differential equations. A feature of the approach is that we iterate between full order solvers with new parameters to compute a POD basis function and sensitivity-based parameter estimation for the ROM problems. The algorithm is applied using neuroimaging data for two clinical test cases, and we can demonstrate that the reduced-order approach drastically decreases the computational effort.     

A model for HCMV infection in immunosuppressed patients

We propose a model for HCMV infection in healthy and immunosuppressed patients. First, we present the biological model and formulate a system of ordinary differential equations to describe the pathogenesis of primary HCMV infection in immunocompetent and immunosuppressed individuals. We then investigate how clinical data can be applied to this model. Approximate parameter values for the model are derived from data available in the literature and from mathematical and physiological considerations. Simulations with the approximated parameter values demonstrates that the model is capable of describing primary, latent, and secondary (reactivated) HCMV infection. Reactivation simulations with this model provide a window into the dynamics of HCMV infection in (D-R+) transplant situations, where latently-infected recipients (R+) receive transplant tissue from HCMV-naive donors (D-).     

An existence result for nonisothermal immiscible incompressible 2‐phase flow in heterogeneous porous media

In the present article, we study the temperature effects on two‐phase immiscible incompressible flow through a porous medium. The mathematical model is given by a coupled system of 2‐phase flow equations and an energy balance equation. The model consists of the usual equations derived from the mass conservation of both fluids along with the Darcy‐Muskat and the capillary pressure laws. The problem is written in terms of the phase formulation; ie, the saturation of one phase, the pressure of the second phase, and the temperature are primary unknowns. The major difficulties related to this model are in the nonlinear degenerate structure of the equations, as well as in the coupling in the system. Under some realistic assumptions on the data, we show the existence of weak solutions with the help of an appropriate regularization and a time discretization. We use suitable test functions to obtain a priori estimates. We prove a new compactness result to pass to the limit in nonlinear terms.     

Simulation of nonlinear fractional dynamics arising in the modeling of cognitive decision making using a new fractional neural network

By the rapid growth of available data, providing data-driven solutions for nonlinear (fractional) dynamical systems becomes more important than before. In this paper, a new fractional neural network model that uses fractional order of Jacobi functions as its activation functions for one of the hidden layers is proposed to approximate the solution of fractional differential equations and fractional partial differential equations arising from mathematical modeling of cognitive-decision-making processes and several other scientific subjects. This neural network uses roots of Jacobi polynomials as the training dataset, and the Levenberg-Marquardt algorithm is chosen as the optimizer. The linear and nonlinear fractional dynamics are considered as test examples showing the effectiveness and applicability of the proposed neural network. The numerical results are compared with the obtained results of some other networks and numerical approaches such as meshless methods. Numerical experiments are presented confirming that the proposed model is accurate, fast, and feasible.     

A differential equation model of North American cinematic box-office dynamics

A new mathematical model is presented for the box-office dynamicsof a motion picture released in North America. Though previouswork on this problem has usually involved probabilistic methods,the new model for describing cumulative box-office gross usesa continuous-time, differential-equation approach. This model,which consists of a system of three nonlinear, coupled, ordinarydifferential equations, incorporates the effects of marketingand advertising expenses, audience reaction, critical reviews,and previous box-office behavior, among other factors. Analyticalasymptotic results are presented for various parameter regimes.In the general case, the model must be solved numerically. Numericalsimulations are tested against actual revenue data from severalrecent movies to analyse the model's accuracy. An algorithmfor practical usage of the model is presented.     

Homogenization results for a coupled system modelling immiscible compressible two-phase flow in porous media by the concept of global pressure

A model for immiscible compressible two-phase flow in heterogeneous porous media is considered. Such models appear in gas migration through engineered and geological barriers for a deep repository for radioactive waste. The main feature of this model is the introduction of a new global pressure and it is fully equivalent to the original equations. The resulting equations are written in a fractional flow formulation and lead to a coupled degenerate system which consists of a nonlinear parabolic (the global pressure) equation and a nonlinear diffusion–convection one (the saturation) equation with rapidly oscillating porosity function and absolute permeability tensor. The major difficulties related to this model are in the nonlinear degenerate structure of the equations, as well as in the coupling in the system. Under some realistic assumptions on the data, we obtain a nonlinear homogenized problem with effective coefficients which are computed via a cell problem and give a rigorous mathematical derivation of the upscaled model by means of two-scale convergence.     

Analysis and Validation of a Mathematical Model for Intracranial Pressure Dynamics

Lumped parameter, compartmental models provide a promising Method for mathematically studying the dynamics of human intracranial pressure. In this modeling approach, a system of fully time-dependent differential equations for interacting compartmental pressures is obtained by considering the intracranial system to be confined within the almost-rigid skull and developing continuity equations associated with conservation of mass. Intracranial volumes and flows are related to compartmental pressure differences through compliance and resistance parameters. In the nonlinear case where compliances are not constant, there is a lack of physical information about these parameters. Consequently, it is vital that any mathematical model with an assumed pressure-dependent compliance be validated through comparison with experimental data. The present work develops a logistic representation for the compliance between the cerebrospinal fluid and brain matter compartments. The nonlinear mathematical model involving this logistic compliance is validated here by comparing its predicted response for bolus injections of cerebrospinal fluid to laboratory data generated in an animal model. Comparison with the animal studies fully supports the validity of the mathematical model with the logistic compliance.     

S-curve regression model in fuzzy environment

The purpose of this paper is to discuss the problem for least squares fitting of fuzzy-valued data, which are expressed as fuzzy numbers, and to develop an S-shaped curve regression model for fitting this type of data. It is shown that the solution of the S-curve regression model is equivalent to the solution of the corresponding linear equations, and, furthermore, the solution can be explicitly obtained by solving the linear equations.     

Joint modeling of longitudinal proportional measurements and survival time with a cure fraction

In cancer clinical trials and other medical studies, both longitudinal measurements and data on a time to an event (survival time) are often collected from the same patients. Joint analyses of these data would improve the efficiency of the statistical inferences. We propose a new joint model for the longitudinal proportional measurements which are restricted in a finite interval and survival times with a potential cure fraction. A penalized joint likelihood is derived based on the Laplace approximation and a semiparametric procedure based on this likelihood is developed to estimate the parameters in the joint model. A simulation study is performed to evaluate the statistical properties of the proposed procedures. The proposed model is applied to data from a clinical trial on early breast cancer.