An energetically coherent lumped parameter model of the left ventricle specially developed for educational purposes
Introduction
The study of the cardiovascular system is strongly multidisciplinary, bringing together scientists in very different research areas. Interdisciplinary teams provide the key to answering questions concerning the many diverse forms of biological interaction among organs, between these organs and their environment and their reactions under physiological conditions or specific inputs.
Taking into account this complexity, if the modeller wants to create a model for educational purposes, he/she will certainly need to keep the model as simple and intuitive as possible, forgetting sometimes his theoretically rigorous formulations, to try new ones, only justifiable a posteriori. The idea is not to offer an “arcanum of mathematical expressions that have little meaning to the ordinary physiologist who is not strongly motivated to struggle with the inadequacies of outmoded ways of thinking” [1], but a comprehensive model, energetically coherent, relatively easy and fast to simulate, that may simulate sufficiently realistically the main experiences done by physiologists to understand ventricular dynamics. A modular approach to dynamic systems modelling is particularly useful for integrating complicated models and their numerous interactions [2]. One of these methodologies, called “Bond Graphs” has found extensive application in a broad variety of applications [3]. Nevertheless, they have been used only sparingly in the field of biomedical engineering. Few Bond Graph-based models of the ventricular dynamics have been published, despite the suitability of this system for a modular system modelling approach and the fact that its internal structure is ideally suited to build multi-physical models for simulation.
In this paper a simple model of the left ventricle (LV) of the heart for educational purposes will be presented step by step: the system and the physiological experimental set-ups utilized to characterize the ventricle will be presented first; then the LV will be presented as a set of sub-systems, going from phenomenological models to more sophisticated representations in terms of basic mechanisms. Then, a variety of “test cases”, which can be useful to characterize the ventricle, will be studied. Finally, the conclusions and perspectives of this work will be presented.
Section snippets
The isolated LV preparation
With its complex anatomy and millions of intricate mechanisms and interactions, the heart is such a complex system that the development of a detailed model is a major challenge needing sophisticated mathematical approaches and numerical techniques [4].
To deal with such a complexity, physiologists have developed chirurgical isolation techniques to separate functionally the LV from the rest of the organ. In the body, the heart is submitted to many external influences: variable conditions in the
A brief history of simple ventricular models: the dynamic compliance (DC)
The model shown in Fig. 2a gives a very simple and well-known representation of the LV behaviour. In its simplest representation, the LV is described as “an elastic muscular bag contracting and relaxing periodically”, mathematically represented by a “dynamic compliance” element, varying with time, with a period equal to the time of a beat, as it is shown in Fig. 2a. In electricity, capacitors are energy-storage elements, being filled with current flow until reaching saturation. The same
More realistic modelling of the LV: from the mechanisms of contraction to hemodynamics
Reality is obviously much more complex than the simple model briefly described in Section 3. In this section we will try to develop a model for understanding and describing the cardiac dynamics of the ventricle starting from the molecular level up to its hemodynamics.
Fig. 3 shows different anatomical sub-levels of the ventricle that will be taken into account for a “more realistic” model of the LV. Starting from the mechanisms of contraction in the cardiac muscle at the contractile proteins
A Bond Graph model of the LV
When working in multi-domain models, important aspects such as kind of quantities, (e.g type of compounds, pressure, flow) measurement units and model consistency must be taken into account [15]. In order to guarantee the intrinsic validity of the final mathematical model, the translation of model specifications into equations must be based on general principles, such as mass or energy conservation according to the different domains of represented physical processes [13]. The implementation of
Simulation results
In this section, some simulation results are presented for different configurations of the model (case studies).
Reference case: physiological conditions
Fig. 8a illustrates several superposed cycles starting in a transient state and then arriving to a periodic state. The transient phase is due to our choice of initial conditions (starting blood volumes in the aorta and the LV are small). Obviously, no real transient would be that large but this figure demonstrates the amplitude of potential
Conclusions and perspectives
Despite many advances in numerical techniques and calculation power, modelling of biological systems, even in isolation, represents a real human and computational challenge; indeed, biological systems are complex and not sufficiently known.
In the last 10 years, functional cardiovascular imaging and modelling have known many impressive advances. For instance, it is becoming increasingly possible to obtain good dynamic images of ventricular pumping or of the mechanical and haemodynamic behaviour
Summary
We present a Bond Graph model and simulation of the cardiac dynamics of the left ventricle. It models all the levels from the mechanisms of contraction up to the hemodynamics of the left ventricle (LV). In particular, we show a detailed explanation of the different equations used for each part of the model, clearly distinguishing the different energetic levels involved: a chemical reaction of the contractile protein represented by a resistance (R element), the chemo-mechanical storage and
Vanessa Díaz-Zuccarini graduated in Mechanical Engineering from Universidad Simón Bolívar (Venezuela). She moved to France where she obtained her PhD in Automatic Control and Industrial Informatics (Modelling and Simulation in Cardiac Dynamics) with “Félicitations du Jury” (First Class Honors) at the Ecole Centrale de Lille in 2003. She was awarded a Marie Curie Intra-European Fellowship in 2004 and is currently a Marie Curie Research Fellow in the Department of Medical Physics at the
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Cited by (14)
An in silico case study of idiopathic dilated cardiomyopathy via a multi-scale model of the cardiovascular system
2014, Computers in Biology and MedicineCitation Excerpt :The same equations as in the MSM of the LV, are used to describe the right ventricle with a lower maximum characteristic elastance. The maximum characteristic elastance of the right ventricle (1.75 mmHg/ml) was chosen in order to obtain the desired haemodynamical values in the model, whereas the elastance of 5 mmHg/ml for the left ventricle originates from Díaz-Zuccarini et al. [13]. An overview of the scales described in the previous sub-sections and the parameters considered to interlink them is presented in [18].
A multi-physics and multi-scale lumped parameter model of cardiac contraction of the left ventricle: A conceptual model from the protein to the organ scale
2012, Computers in Biology and MedicineCitation Excerpt :The coupling of this model within MATLAB® was completed by connecting the different scales with a driving function. Each scale was validated separately based on [34,35,42]. Whilst, the protein domain reacts in the responsiveness to activating Ca2+ ion from the cellular level, particularly, the protein level distinguishes on how molecular kinetics assess the macroscopic (organ) domain.
Systemic modelling and computational physiology: The application of Bond Graph boundary conditions for 3D cardiovascular models
2009, Simulation Modelling Practice and TheoryCitation Excerpt :Again this is in strong contrast with, but complementary to, the detailed approach of computational modelling. More sophisticated approaches describing the X-bridges mechanism (actin-myosin chemical reaction) in the sarcomere, describing the regulation of mechanical activity in the intact cardiac muscle, considering the effects of the free calcium transients and the mechanical constraints, and emphasizing the role of the troponin complex in regulating muscle activity or developing more detailed single ventricular cell models that simulate action potentials, Ca2+-handling mechanisms, and isometric force generation by the myofilaments are available [15,16]. However, these models were considered too complicated at this stage; thus, a simple representation of the actin-myosin mechanism was chosen.
Energy-based advection modelling using bond graphs
2022, Journal of the Royal Society InterfaceLumped parameter heart model with valve dynamics
2019, Russian Journal of Numerical Analysis and Mathematical Modelling
Vanessa Díaz-Zuccarini graduated in Mechanical Engineering from Universidad Simón Bolívar (Venezuela). She moved to France where she obtained her PhD in Automatic Control and Industrial Informatics (Modelling and Simulation in Cardiac Dynamics) with “Félicitations du Jury” (First Class Honors) at the Ecole Centrale de Lille in 2003. She was awarded a Marie Curie Intra-European Fellowship in 2004 and is currently a Marie Curie Research Fellow in the Department of Medical Physics at the University of Sheffield, where she is carrying out research on the coupling of 3D models with lumped parameter models for cardiovascular applications. Dr. Vanessa Díaz-Zuccarini is the National Coordinator of the UK Group of the Marie Curie Fellows Association and is a member of the European Society of Biomechanics.
Jacques LeFèvre (Born on 29-02-1944). After obtaining his B.Sc. and M.Sc. degrees in engineering, Jacques LeFvre got in 1979 a Ph.D. in Applied Mathematics and in Bioengineering from Louvain University (Belgium). Since then, he has worked in simulation methodology, mathematical modelling and cardiovascular physiology in several universities: Louvain, Kings College Medical School (London), Technion (Haifa) and U. of Pennsylvania (Philadelphia). He was a visiting professor in the Universities of Pennsylvania, Athens, City University (London), Padova and Lyon. He is currently an independent consultant in modelling for IDEA.SIM LTD, a company which he started in 1998 in London; he is also a part time professor in Ecole Centrale de Lille, a French Grande Ecole dIngnieurs. In addition to cardiovascular modelling, his research deals also with neuro-fuzzy-genetic algorithms and socio-environmental modelling. He originated the use of fractals and of bond graphs in cardiovascular modelling (1979, 1982). Since 1996, he has been developed a new method of kinetic lumped modelling termed kinetic graphs which has wide applications not only in cellular physiology and biochemistry but also in socio-economical and environmental modelling. He has published widely on all these subjects. He is also an avid solo caver and part time professional guide in alpine caving and canyoning.