Applied Mathematical Models in Human Physiology
Pernille Adeler
Johnny T Ottesen2004
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Abstract
The purpose of this book is to study mathematical models of human physiology. The book is a result of work by Math-Tech (in Copenhagen, Denmark) and the BioMath group at the Department of Mathematics and Physics at Roskilde University (in Roskilde, Denmark) on mathematical models related to anesthesia simulation. The work presented in this book has been carried out as part of a larger project SIMA (SIMulation in Anesthesia) 1 , which has resulted in the production of a commercially available anesthesia simulator and several scientific research publications contributing to the understanding of human physiology. This book contains the scientific contributions and does not discuss the details of the models implemented in the SIMA project.
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MATHEMATICAL MODELING OF CARDIAC BLOOD FLOW IN HUMANS
Lumped parameter model is a very useful type of mathematical modelling where the physical system is made analogous to an electrical network. Lumped parameter model is represented graphically by a circuit diagram in which vertices represent the voltages and the edges the current in the circuit. The mathematical analysis of such a circuit model is very convenient and eases the analysis of the actual physical system. These models are constructed by building the analogy between cardiovascular system and electrical circuit.This analogy makes use of simple ordinary differential equations in time which can be solved either numerically or analytically. A mathematical model to model the blood pressure variation in the four chambers of heart at representative altitudes of h= 2 km, h = 4 km and h = 5 kmusing lumped compartments of blood circulation is presented in this paper. This lumped parameter model consists of eight compartments that include the pumping heart, the systemic circulation and the pulmonary circulation. The governing equations for pressure and flow in each compartment are derived from the following three equations: Ohm's law, conservation of volume and the definition of compliances. The g-factor is added to the model in order to accommodate for the effects due to gravitation.
An integrated mathematical model of the cardiovascular and respiratory systems
International Journal for Numerical Methods in Biomedical Engineering, 2015
SummaryThis study presents a lumped model for the human cardiorespiratory system. Specifically, we incorporate a sophisticated gas dissociation and transport system to a fully integrated cardiovascular and pulmonary model. The model provides physiologically consistent predictions in terms of hemodynamic variables such as pressure, flow rate, gas partial pressures, and pH. We perform numerical simulations to evaluate the behavior of the partial pressures of oxygen and carbon dioxide in different vascular and pulmonary compartments. For this, we design the rest condition with low oxygen requirements and carbon dioxide production and exercise conditions with high oxygen demand and carbon dioxide production. Furthermore, model sensitivity to more relevant model parameters is studied. Copyright © 2015 John Wiley & Sons, Ltd.
An integrated mathematical model of the human cardiopulmonary system: model development
American Journal of Physiology-Heart and Circulatory Physiology, 2015
Several cardiovascular and pulmonary models have been proposed in the last few decades. However, very few have addressed the interactions between these two systems. Our group has developed an integrated cardiopulmonary model (CP Model) that mathematically describes the interactions between the cardiovascular and respiratory systems, along with their main short-term control mechanisms. The model has been compared with human and animal data taken from published literature. Due to the volume of the work, the paper is divided in two parts. The present paper is on model development and normophysiology, whereas the second is on the model's validation on hypoxic and hypercapnic conditions. The CP Model incorporates cardiovascular circulation, respiratory mechanics, tissue and alveolar gas exchange, as well as short-term neural control mechanisms acting on both the cardiovascular and the respiratory functions. The model is able to simulate physiological variables typically observed in a...
[Mathematical modelling in physiology]
Rossiĭskii fiziologicheskiĭ zhurnal imeni I.M. Sechenova / Rossiĭskaia akademiia nauk
The article illustrates the method of mathematical modelling in physiology as a unique tool to study physiological processes. A number of demonstrated examples appear as a result of long-term experience in mathematical modelling of electrical and mechanical phenomena in the heart muscle. These examples are presented here to show that the modelling provides insight into mechanisms underlying these phenomena and is capable to predict new ones that were previously unknown. While potentialities of the mathematical modelling are analyzed with regard to the myocardium, they are quite universal to deal with any physiological processes.
Development of a Numerical Simulation Model of the Cardiovascular System
Artificial Organs, 2008
It is the goal of this section to publish material that provides information regarding specific issues, aspects of artificial organ application, approach, philosophy, suggestions, andlor thoughts for the future.
Engineering Modeling of Human Cardiovascular System
Cardiovascular system is primarily considered as the human body's transport system. Oxygen, carbon dioxide, nutrients and other vital substances to the various tissues of human body are carried by the blood which circulates in a closed circulation. The cardiovascular system has been comprised of a combination of several basic compartments, which are structurally connected to and functionally interact with each other. Engineering modeling of such important system has become a useful tool to diagnose the cardiovascular diseases and recommend the appropriate way of their medical treatment. This paper presents a quantified model describing the relationship between the input and output variables of the hemodynamic regulation of the system through implementing a set of first order differential equations that governing this performance and describing its parameters such as pressures, volumes and flows in a closed-loop lumped system. Construction of this model was based on the interacti...
Quantitative circulatory physiology: an integrative mathematical model of human physiology for medical education
… in Physiology …, 2007
Abram SR, Hodnett BL, Summers RL, Coleman TG, Hester RL. Quantitative circulatory physiology: an integrative mathematical model of human physiology for medical education. We have developed Quantitative Circulatory Physiology (QCP), a mathematical model of integrative human physiology containing over 4,000 variables of biological interactions. This model provides a teaching environment that mimics clinical problems encountered in the practice of medicine. The model structure is based on documented physiological responses within peer-reviewed literature and serves as a dynamic compendium of physiological knowledge. The model is solved using a desktop, Windows-based program, allowing students to calculate time-dependent solutions and interactively alter over 750 parameters that modify physiological function. The model can be used to understand proposed mechanisms of physiological function and the interactions among physiological variables that may not be otherwise intuitively evident.
A new hybrid (hydro-numerical) model of the circulatory system
Bulletin of the Polish Academy of Sciences: Technical Sciences, 2013
The paper presents a hybrid (hydro-numerical) circulatory model built to be used as a complementary tool for clinical purposes. It was developed at the Institute of Biocybernetics and Biomedical Engineering - Polish Academy of Sciences (Poland) in co-operation with the Institute of Clinical Physiology - National Council of Research (Italy). Main advantages of the model are: 1) high accuracy and repeatability of parameters setting, characteristic of numerical solutions, 2) maximum flexibility achieved by implementing the largest possible number of the model’s elements in the numerical way, 3) ability to test mechanical heart assist devices provided by special computer applications; in the model two physically different signal environments - numerical and hydraulic - are connected by special impedance transformers interfacing physical and numerical parts of the model; 4) eliminating flowmeters, as the voltage controlled flow sources embedded in the system provide information on flows....
Lumped Parameter Model of Cardiovascular-Respiratory Interaction
— The aim of this work was to develop a lumped parameter model of the cardiovascular system and to couple it with a model of respiratory mechanics. In comparison to existing models, modifications and additions have been implemented to include a model of the upper limb vasculature employing the electrical analogy of hemodynamic variables. The model prediction of respiratory effects on arterial pressure was compared with in vivo invasive measurement of blood pressure in patients. The model indicates that the inherent coupling between the cardiovascular and respiratory systems can be described by mathematical relationships of physiological parameters with robust predictions. With specification of parameters based on individual measurements of cardio-respiratory variables, the model can be used in the clinical setting of intensive care units to predict hemodynamic changes and to optimize ventilation and volume loading strategies.
A human cardiopulmonary system model applied to the analysis of the Valsalva maneuver
American Journal of Physiology Heart and Circulatory Physiology, 2001
A human cardiopulmonary system model applied to the analysis of the Valsalva maneuver. Am J Physiol Heart Circ Physiol 281: H2661-H2679, 2001.-Previous models combining the human cardiovascular and pulmonary systems have not addressed their strong dynamic interaction. They are primarily cardiovascular or pulmonary in their orientation and do not permit a full exploration of how the combined cardiopulmonary system responds to large amplitude forcing (e.g., by the Valsalva maneuver). To address this issue, we developed a new model that represents the important components of the cardiopulmonary system and their coupled interaction. Included in the model are descriptions of atrial and ventricular mechanics, hemodynamics of the systemic and pulmonic circulations, baroreflex control of arterial pressure, airway and lung mechanics, and gas transport at the alveolar-capillary membrane. Parameters of this combined model were adjusted to fit nominal data, yielding accurate and realistic pressure, volume, and flow waveforms. With the same set of parameters, the nominal model predicted the hemodynamic responses to the markedly increased intrathoracic (pleural) pressures during the Valsalva maneuver. In summary, this model accurately represents the cardiopulmonary system and can explain how the heart, lung, and autonomic tone interact during the Valsalva maneuver. It is likely that with further refinement it could describe various physiological states and help investigators to better understand the biophysics of cardiopulmonary disease. cardiopulmonary modeling; ventricular interaction; closedloop hemodynamics; baroreflex control; airway mechanics; gas exchange THE DIAGNOSIS AND TREATMENT of cardiopulmonary disease may be improved by using mathematical models of the cardiovascular and pulmonary systems. With this in mind, we developed a model of the cardiopulmonary system of the normal human subject that not only represents the system accurately but also predicts its response to a variety of commonly used diagnostic procedures. To our knowledge, this is the first example of a truly integrative model of the cardiopulmonary system.
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Mathematical modelling of the cardiovascular system
Journal of Engineering Mathematics, 2000
In this paper we will address the problem of developing mathematical models for the numerical simulation of the human circulatory system. In particular, we will focus our attention on the problem of haemodynamics in large human arteries. 2000 Mathematics Subject Classification: 93A30, 35Q30, 74F10, 65N30.
An integrated model of the human cardiopulmonary system
We have developed a mathematical model of the human cardiopulmonary system that is able to simulate the normal functions of the cardiovascular and pulmonary systems, as well as their coupled interactions. Included in the model are descriptions of atrial and ventricular mechanics, the hemodynamics of the systemic and pulmonic circulations, baroreflex control of arterial pressure, airway and lung mechanics, as well as, gas transport at alveolar-capillary membrane. With the suitable parameter set, the integrated cardiopulmonary model yielded pressure, volume and flow waveforms that agree well with published data. In addition, The model demonstrated stability under large amplitude perturbations of the physiological variables, such as Valsalva Maneuver.
Chapter 2 Mathematical Modeling of Physiological Systems
2012
Although mathematical modeling has a long and very rich tradition in physiology, the recent explosion of biological, biomedical, and clinical data from the cellular level all the way to the organismic level promises to require a renewed emphasis on computational physiology, to enable integration and analysis of vast amounts of life-science data. In this introductory chapter, we touch upon four modeling-related themes that are central to a computational approach to physiology, namely simulation, exploration of hypotheses, parameter estimation, and modelorder reduction. In illustrating these themes, we will make reference to the work of others contained in this volume, but will also give examples from our own work on cardiovascular modeling at the systems-physiology level.
Mathematical modelling of flow distribution in the human cardiovascular system
Medical & Biological Engineering & Computing, 1992
Mathematical Modelling of the Cardiovascular System Haemodynamics
2016
In this paper we propose a model of the cardiovascular system, where stressed blood volume is a model parameter instead of an initial condition. Stressed blood volume (SBV) is an important indicator of fluid responsiveness, i.e., this term can help to classify patients between responders and non-responders to fluid therapy. We study a six compartment model, then using the conservation of total blood volume, one differential equation of the original model is omitted and a new model is obtained. Comparing the haemodynamic signals of the previous model and the new version, we show that the simulations are qualitatively similar. One important difference with the original model is that the initial conditions for solving the proposed model are arbitrary. This allows us to perform a sensitivity analysis using automatic differentiation of the reduced model for all model parameters without excluding any parameter a priori, unlike the authors of the original formulation have done. This analysis showed that two parameters, the heart period and the stressed blood volume, have a major effect on the performance of the model.
Modeling of Physiological Flows
MS&A, 2012
The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
Computational Hemodynamic Modeling of the Cardiovascular System
International Journal of System Dynamics Applications, 2014
Computational methods and modeling are widely used in many fields to study the dynamic behaviour of different phenomena. Currently, the use of these models is an accepted practice in the biomedical field. One of the most significant efforts in this direction is applied to the simulation and prediction of pathophysiological conditions that can affect different systems of the human body. In this work, the design and development of a computational model of the human cardiovascular system is proposed. The structure of the model has been built from a physiological base, considering some of the mechanisms associated to the cardiovascular system. Thus, the aim of the model is the prediction, heartbeat by heartbeat, of some hemodynamic variables from the cardiovascular system, in different pathophysiological cardiac situations. A modular approach to development of the model has been considered in order to include new knowledge that could force the model's hemodynamic. The model has been...
Mathematical modelling and numerical simulation of the cardiovascular system
2003
Fig. 2.1. The human circulatory system. The human cardiovascular system has the task of supplying the human organs with blood. Its correct working is obviously crucial and depends on many parameters: external temperature, muscular activity, state of health, just to mention a few. The blood pressure and flow rate then change according to the body needs.
Modeling the cardiovascular system—A mathematical adventure: Part I
2001
Mathematical and numerical investigations of the cardiovascular system, although a relatively new research area, will give rise to some of the major mathematical challenges of the coming decades, the author writes. In this first of two parts, he sketches some of the features that make the human blood circulatory system so challenging to model. The second part of the article (to appear in the next issue of SIAM News) will describe research now under way, including the development of hybrid multiscale models.
Flow Simulations in the Human Cardiovascular System under Variable Conditions
2015
In this paper we study the effects of variations in the external conditions (such as pressure changes due to motion in gravitational field) on the cardiovascular system. The goal is to understand how the flow and pressures propagate in the network in such variable conditions and which are the locations where the effects are significant. This is relevant to the auto regulation mechanism of the pressure and flow, designed to maintain the system in a homeostatic condition.
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