A mathematical model of the systemic circulatory system with logistically defined nervous system regulatory mechanisms

A mathematical model is developed that accurately describes the pressure, volume and flow dynamics of the systemic circulatory system over the full physiological range of human pressures and volumes. At the heart of this model are mathematical representations for the autonomic and central nervous system reflexes which maintain arterial pressure, cardiac output and cerebral blood flow. These representations involve functions in which a maximum effect and a minimum effect are smoothly connected by a logistic transition. A new approach to modelling the pressure – volume relationship in a vessel with smooth muscle contraction is also presented. To test the model, simulations of cardiac arrest and various haemorrhagic situations were conducted, and predicted results were compared with clinical observations. Near-perfect agreement was obtained between predicted and observed values of the mean circulatory filling pressure, cardiac output and arterial pressure decay in the face of significant haemorrhage, and the critical values delineating progressive from non-progressive hypovolaemic shock.     

Global analysis of the initial value problem for a system of O.D.E. modeling the immune system after vaccinations

A model of the human immune system after a series of vaccinations, proposed in [1], is studied, and a global qualitative study of the model in some special cases is proposed. It is shown that all solutions of the model have at most most one maximum at some point t_m, next, for t > t_m, t→ + ∞, monotonically tend to a physiological level, according to the behavior of antibody concentration observed in reality.     

Complex bifurcation/chaotic behavior of acetylcholinesterase and cholineacetyltransferase enzymes system

This investigation utilizes available biokinetic information for the formulation of a diffusion-reaction model in order to simulate the in vivo behavior of acetylcholinesterase (AChE) and cholineacetyltransferase (ChAT) coupled enzymes system. A novel two-enzymes/two-compartments model is used to explore the bifurcation and chaotic behavior of this enzyme system simulating the acetylcholine neurocycle in the brain. Detailed bifurcation analysis over a wide range of parameters is carried out in order to uncover some important information about this system. The results of this investigation relate to the phenomena occurring in the physiological experiments, like periodic stimulation of neural cells and non-regular functioning of acetylcholine receptors. These results can be used to direct more systematic research on cholinergic brain diseases like Alzheimer’s and Parkinson’s diseases as there are strong evidences that these brain disorders are related to concentration of acetylcholine (ACh) in the brain.     

A statistical approach to the determination of stability for dynamical systems modelling physiological processes

One of the questions involved in the formulation of a new model for a physiological phenomenon, when the model represents a dynamical system, is that concerning its qualitative behavior. The determination of the stability of a particular dynamical system is usually made analytically, from a linearization of the system around an equilibrium point. This analytic proof may often be very complex or impossible, leading to the imposition of conditions on the relative magnitude of the structural model parameters or to other partial results. We discuss a general technique whereby a probabilistic judgment is made on the stability of a dynamical system, and we apply it to the study of a particular delay differential system modelling the relationship between insulin secretion and glucose uptake. This technique is applicable in case experimental material is available from which to estimate the dispersion of the model parameters. A stability criterion is obtained via the usual linearization around an equilibrium point, it is approximated as a Taylor series in the parameters truncated after the first term, and its variance is then computed from the dispersion of the parameters. While the conclusion is probabilistic in nature, it can be obtained for a wide class of models and from either sample or individual experimental subject's parameter estimates.     

人体体循环输入阻抗的T-Y型管模型*

本文提出了模拟人体体循环输入阻抗的T－Y型管模型,它是O'Rourke[1]提出的,柳兆荣等[2]发展的非对称T型管模型的改进和推广.我们在非对称T型管基础上加添了代表人体两条腿的倒Y字型分叉管,从而形成了T－Y型模型,同时将模型中所有分支管都处理成具有纵向约束的粘弹性管,上肢终端小动脉床采用弹性腔模型,下肢终端小动脉床则采用传统的纯阻力形式.选取了适当参数值后,计算了正常生理情形下和高血压病理情形下的人体体循环的输入阻抗.结果显示理论模拟值和实测结果无论是生理情形或是病理情形都有良好的符合程度.T－Y型管模型比非对称T型管模型更接近人体体循环的生理结构.同时又比过份复杂的多重分叉管模型简单明了得多,因此在考察各种参数对升主动脉输入阴抗的影响.研究心室和血管的耦合机制中将是一个有实用价值的模型.     

A complete steady state model of solute and water transport in the kidney

The purpose of this paper is to incorporate a detailed model, along with an optimized set of parameters for the proximal tubule, into J. L. Stephenson's current central core model of the nephron. In this model a set of equations for the proximal tubule are combined with Stephenson's equations for the remaining four tubules and interstitium, to form a complete nonlinear system of 34 ordinary differential and algebraic equations governing fluid and solute flow in the kidney. These equations are then discretized by the Crank-Nicholson scheme to form an algebraic system of nonlinear equations for the unknown concentrations, flows, hydrostatic pressure, and potentials. The resulting system is solved via factored secant update with a finite-difference approximation to the Jacobian. Finally, numerical simulations performed on the model showed that the modeled behavior approximates, in a general way, the physiological mechanisms of solvent and solute flow in the kidney.     

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-).     

A mathematical model of immune competition related to cancer dynamics

This paper deals with the qualitative analysis of a model describing the competition among cell populations, each of them expressing a peculiar cooperating and organizing behavior. The mathematical framework in which the model has been developed is the kinetic theory for active particles. The main result of this paper is concerned with the analysis of the asymptotic behavior of the solutions. We prove that, if we are in the case when the only equilibrium solution if the trivial one, the system evolves in such a way that the immune system, after being activated, goes back toward a physiological situation while the tumor cells evolve as a sort of progressing travelling waves characterizing a typical equilibrium/latent situation. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical bifurcation control of Mackey–Glass system

In this paper, a mathematical physiological model, Mackey–Glass system of a delay differential equation, is considered. With a greater delay, a periodic solution arises, which characterizes the disease of chronic granulocytic leukemia (CGL). To treat such disease, a blood transfusion feedback control is considered, from the point of view of mathematical control. By using a nonstandard finite-difference (NSFD) scheme to the control system, we obtain a numerical discrete system and analyze its Neimark–Sacker and fold bifurcations. The results imply that the condition of the illness could be relieved by transfusing blood to the patient, if the control is a delay control. Finally, the effectiveness of the control are illustrated by several numerical simulations.     

Inverse problems related to ion channels

Ion channels are proteins with a hole down theirmiddle that allow ions to move across otherwise impermeable cellmembranes, thereby controlling many important physiological functions. The transport process of the ions can be described using the Poisson-Nernst-Plank equations, a system of coupled nonlinear partial differential equations. Based on this model we derive a simplified surrogate model that captures the main features of the associated current-voltage curves of the ion channel. This surrogate model is then used to identify individual channel parameters based on current data. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

人体体动脉脉搏波和输入阻抗的计算机模拟

本文通过对具有连续出流的分支、锥削管的一维非线性瞬态流分析,研究了人体体动脉中压力和流量脉搏波的传输和血管输入阻抗的变化规律.通过耦合支配着动脉中脉搏波传播的连续性方程和运动方程,文章导得了一组可以用特征线法求解的拟线性双曲型偏微分方程组.同时,对血管系统中常见的几种边界条件进行了适当的处理.文章利用血管几何参数和物理参数的生理实验数据,对血管系统的脉搏波传播和输入阻抗进行了计算机模拟.结果表明,从本文提出的分析模型得到的压力和流量波形以及输入阻抗与有关文献公布的实验结果吻合得相当好.     

生理药动学模型参数估计问题

根据生理药动学模型的特点,把非线性药动学模型转化为线性模型,并验证线性模型的精确性.在此基础上,构造估计药动学模型参数的目标函数,并利用非线性优化算法求解模型参数.仿真结果表明,我们的算法具有快速、精确和稳定的特点.给出了一种快速估计复杂生理药动学模型参数的方法,这为解决复杂生理药动学模型的参数估计问题提供了一种有效工具.     

Three-neuron model of compensatory eye movements in response to vestibular stimulation

Generation of compensatory eye movements which arise in response to vestibular stimulation is described in the framework of a three-neuron pathway, by which the vestibulo-ocular reflex is conducted. A mathematical model of the reflex based on the literature is presented in the paper. All parameters and variables of the model proposed have a clear physiological meaning. Incompleteness of the vestibulo-ocular interconnections described in the literature is shown. For the particular case of head rotation about the vertical axis, an experiment is proposed and performed; the results of the experiment allow one to identify unknown parameters of the model. The torsional vestibulo-ocular reflex readaptation process to usual earth gravity conditions after long-term orbital flight was experimentally investigated. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 8, pp. 175–193, 2005.     

Local Compliance Effects on the Global Pressure-Volume Relationship in Models of Intracranial Pressure Dynamics

The experimentally-measured pressure-volume relationship for the human intracranial system is a nonlinear ‘S-shaped’ curve with two pressure plateaus, a point of inflection, and a vertical asymptote at high pressures where all capacity for volume compensation is lost. In lumped-parameter mathematical models of the intracranial system, local compliance parameters relate volume adjustments to dynamic changes in pressure differences between adjacent model subunits. This work explores the relationship between the forms used for local model compliances and the calculated global pressure-volume relationship. It is shown that the experimentally-measured global relationship can be recovered using physiologically motivated expressions for the local compliances at the interfaces between the venous-cerebrospinal fluid (CSF) subunits and arterial-CSF subunits in the model. Establishment of a consistent link between local model compliances and the physiological bulk pressure-volume relationship is essential if lumped-parameter models are to be capable of realistically predicting intracranial pressure dynamics.     

Coupling 3D and 1D Skeletal Muscle Models

This work introduces a modelling framework towards a forward dynamics simulation of skeletal muscle mechanics that couples three-dimensional (3D) continuum-mechanical-based Finite Element (FE) simulations to rigid body simulations. In this regard, this is a methodological approach, which incorporates different methods to realise simulations of the musculoskeletal system. Such simulations are at present computationally not feasible. To set up such a modelling framework the upper limp is selected. Here, the upper limb consists of an antagonistic muscle pair, the elbow (a simple hinge joint) and an external load. The skeletal muscles are represented by a 3D continuum-mechanical model. The tendons are, for now, assumed to be rigid. The results demonstrate the ability of the system to converge to a physiological realistic position. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis and control of the bifurcation of Hodgkin–Huxley model

The Hodgkin–Huxley (HH) equations are parameterized by a number of parameters and show a variety of qualitatively different behaviors depending on the various parameters. This paper finds the bifurcation would occur when the leakage conductance g_l is lower than a special value. The Hopf bifurcation of HH model is controlled by applying a simple and unified state-feedback method and the bifurcation point is moved to an unreachable physiological point at the same time, so in this way an absolute bifurcation control is achieved. The simulation results demonstrate the validity of such theoretic analysis and control method. This new method could be a great help to the design of new closed-loop electrical stimulation systems for patients suffering from different nerve system dysfunctions.     

Robust Adaptive Identification of Fuzzy Systems with Uncertain Data

This study presents a method of adaptive identification of parameters describing Sugeno fuzzy inference system in presence of bounded disturbances while maintaining the readability and interpretability of the fuzzy model during and after identification. This method do not require any a priori knowledge of a bound on the disturbance and noise and of a bound on the unknown parameters values. The method can be used for the robust and adaptive identification of slowly time varying nonlinear systems using fuzzy inference systems. The suggested method was used to build a fuzzy expert system that approximates the functional relationship between physical fitness and some of the measurable physiological parameters by their real measurements and opinion (human-experiences) of a medical expert.     

A mathematical approach to HIV infection dynamics

In order to obtain a comprehensive form of mathematical models describing nonlinear phenomena such as HIV infection process and AIDS disease progression, it is efficient to introduce a general class of time-dependent evolution equations in such a way that the associated nonlinear operator is decomposed into the sum of a differential operator and a perturbation which is nonlinear in general and also satisfies no global continuity condition. An attempt is then made to combine the implicit approach (usually adapted for convective diffusion operators) and explicit approach (more suited to treat continuous-type operators representing various physiological interactions), resulting in a semi-implicit product formula. Decomposing the operators in this way and considering their individual properties, it is seen that approximation–solvability of the original model is verified under suitable conditions. Once appropriate terms are formulated to describe treatment by antiretroviral therapy, the time-dependence of the reaction terms appears, and such product formula is useful for generating approximate numerical solutions to the governing equations. With this knowledge, a continuous model for HIV disease progression is formulated and physiological interpretations are provided. The abstract theory is then applied to show existence of unique solutions to the continuous model describing the behavior of the HIV virus in the human body and its reaction to treatment by antiretroviral therapy. The product formula suggests appropriate discrete models describing the dynamics of host pathogen interactions with HIV1 and is applied to perform numerical simulations based on the model of the HIV infection process and disease progression. Finally, the results of our numerical simulations are visualized and it is observed that our results agree with medical and physiological aspects.     

A load share model for non-identical components of a k-out-of-m system

We consider a k-out-of-m load sharing system when the lifetimes of the components are not necessarily identically distributed random variables. For such systems, a model for the load sharing phenomenon through the exponentiated conditional survival functions of ordered failure times is proposed. This model is more general than the load sharing model with identically distributed component lifetimes and leads to a different family of distributions for ordered random variables. A general expression for the reliability of the system is given. The computations of the reliability for a two component parallel load sharing system corresponding to the exponential and Weibull distributions are discussed. For illustrative purpose, we discuss the inference procedures for a two component parallel load sharing system corresponding to the exponential distributions. A simulation study is carried out to assess the proposed estimation and testing procedures. The applicability of the proposed load sharing model is shown through two data sets.