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Mathematics
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Development and Applications of Multi-Scale Mathematical Models in Cardiology
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Development and Applications of Multi-Scale Mathematical Models in Cardiology

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematics and Computer Science".

Deadline for manuscript submissions: closed (31 January 2022) | Viewed by 23923

Special Issue Editor

Dr. Rafael Sebastian

E-Mail Website
Guest Editor
Computational Multiscale Simulation Lab (CoMMLab), Escola Tècnica Superior d’Enginyeria (ETSE-UV), Universitat de València, 46010 València, Spain
Interests: biophysical modeling; parallel computing; data assimilation; multi-scale modeling; model personalization; image and signal processing; mathematical modeling of anatomical structures; digital twin

Special Issue Information

Dear Colleagues,

Over the last years different types of mathematical models have proven to be able to capture the complex dynamic behavior of the heart at different scales, from the cell to the organ level. There are different approaches depending on the final goal, from detailed biophysical models that can account on subcellular details, allowing for instance to model the effect of drugs, to phenomenological models that can reproduce the macroscopic behavior of the heart and help in decision-making tasks. Both types of models usually take into account the cellular, tissue and organ levels separately and with different levels of detail.

The purpose of this special issue is to establish the current state-of-the art in mathematical modeling of the heart that can lead to the digital twin concept, taking into account the final purpose of each type of model, as well as their advantages and applications in medicine.  Of special interest are research papers that deal with multi-scale biophysical models, mathematical models that are validated for translation into real clinical environments, and simplified phenomenological models that are accurate and can deliver results in times compatible with clinical environments.

Dr. Rafael Sebastian
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Mathematical modeling of the heart
  • Biophysical models of myocytes
  • Electro-mechanical coupling of heart cells
  • Phenomenological heart models
  • Models based on cellular automata

Published Papers (6 papers)

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Research

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21 pages, 6745 KiB  
Article
An Automata-Based Cardiac Electrophysiology Simulator to Assess Arrhythmia Inducibility
by Dolors Serra, Pau Romero, Ignacio Garcia-Fernandez, Miguel Lozano, Alejandro Liberos, Miguel Rodrigo, Alfonso Bueno-Orovio, Antonio Berruezo and Rafael Sebastian
Mathematics 2022, 10(8), 1293; https://0-doi-org.brum.beds.ac.uk/10.3390/math10081293 - 13 Apr 2022
Cited by 11 | Viewed by 2434
Abstract
Personalized cardiac electrophysiology simulations have demonstrated great potential to study cardiac arrhythmias and help in therapy planning of radio-frequency ablation. Its application to analyze vulnerability to ventricular tachycardia and sudden cardiac death in infarcted patients has been recently explored. However, the detailed multi-scale [...] Read more.
Personalized cardiac electrophysiology simulations have demonstrated great potential to study cardiac arrhythmias and help in therapy planning of radio-frequency ablation. Its application to analyze vulnerability to ventricular tachycardia and sudden cardiac death in infarcted patients has been recently explored. However, the detailed multi-scale biophysical simulations used in these studies are very demanding in terms of memory and computational resources, which prevents their clinical translation. In this work, we present a fast phenomenological system based on cellular automata (CA) to simulate personalized cardiac electrophysiology. The system is trained on biophysical simulations to reproduce cellular and tissue dynamics in healthy and pathological conditions, including action potential restitution, conduction velocity restitution and cell safety factor. We show that a full ventricular simulation can be performed in the order of seconds, emulate the results of a biophysical simulation and reproduce a patient’s ventricular tachycardia in a model that includes a heterogeneous scar region. The system could be used to study the risk of arrhythmia in infarcted patients for a large number of scenarios. Full article
(This article belongs to the Special Issue Development and Applications of Multi-Scale Mathematical Models in Cardiology)
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35 pages, 11516 KiB  
Article
An Integrated Workflow for Building Digital Twins of Cardiac Electromechanics—A Multi-Fidelity Approach for Personalising Active Mechanics
by Alexander Jung, Matthias A. F. Gsell, Christoph M. Augustin and Gernot Plank
Mathematics 2022, 10(5), 823; https://0-doi-org.brum.beds.ac.uk/10.3390/math10050823 - 04 Mar 2022
Cited by 11 | Viewed by 2733
Abstract
Personalised computer models of cardiac function, referred to as cardiac digital twins, are envisioned to play an important role in clinical precision therapies of cardiovascular diseases. A major obstacle hampering clinical translation involves the significant computational costs involved in the personalisation of biophysically [...] Read more.
Personalised computer models of cardiac function, referred to as cardiac digital twins, are envisioned to play an important role in clinical precision therapies of cardiovascular diseases. A major obstacle hampering clinical translation involves the significant computational costs involved in the personalisation of biophysically detailed mechanistic models that require the identification of high-dimensional parameter vectors. An important aspect to identify in electromechanics (EM) models are active mechanics parameters that govern cardiac contraction and relaxation. In this study, we present a novel, fully automated, and efficient approach for personalising biophysically detailed active mechanics models using a two-step multi-fidelity solution. In the first step, active mechanical behaviour in a given 3D EM model is represented by a purely phenomenological, low-fidelity model, which is personalised at the organ scale by calibration to clinical cavity pressure data. Then, in the second step, median traces of nodal cellular active stress, intracellular calcium concentration, and fibre stretch are generated and utilised to personalise the desired high-fidelity model at the cellular scale using a 0D model of cardiac EM. Our novel approach was tested on a cohort of seven human left ventricular (LV) EM models, created from patients treated for aortic coarctation (CoA). Goodness of fit, computational cost, and robustness of the algorithm against uncertainty in the clinical data and variations of initial guesses were evaluated. We demonstrate that our multi-fidelity approach facilitates the personalisation of a biophysically detailed active stress model within only a few (2 to 4) expensive 3D organ-scale simulations—a computational effort compatible with clinical model applications. Full article
(This article belongs to the Special Issue Development and Applications of Multi-Scale Mathematical Models in Cardiology)
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19 pages, 13792 KiB  
Article
Sensitivity Analysis of In Silico Fluid Simulations to Predict Thrombus Formation after Left Atrial Appendage Occlusion
by Jordi Mill, Victor Agudelo, Andy L. Olivares, Maria Isabel Pons, Etelvino Silva, Marta Nuñez-Garcia, Xabier Morales, Dabit Arzamendi, Xavier Freixa, Jérôme Noailly and Oscar Camara
Mathematics 2021, 9(18), 2304; https://0-doi-org.brum.beds.ac.uk/10.3390/math9182304 - 18 Sep 2021
Cited by 27 | Viewed by 3883
Abstract
Atrial fibrillation (AF) is nowadays the most common human arrhythmia and it is considered a marker of an increased risk of embolic stroke. It is known that 99% of AF-related thrombi are generated in the left atrial appendage (LAA), an anatomical structure located [...] Read more.
Atrial fibrillation (AF) is nowadays the most common human arrhythmia and it is considered a marker of an increased risk of embolic stroke. It is known that 99% of AF-related thrombi are generated in the left atrial appendage (LAA), an anatomical structure located within the left atrium (LA). Left atrial appendage occlusion (LAAO) has become a good alternative for nonvalvular AF patients with contraindications to anticoagulants. However, there is a non-negligible number of device-related thrombus (DRT) events, created next to the device surface. In silico fluid simulations can be a powerful tool to better understand the relation between LA anatomy, haemodynamics, and the process of thrombus formation. Despite the increasing literature in LA fluid modelling, a consensus has not been reached yet in the community on the optimal modelling choices and boundary conditions for generating realistic simulations. In this line, we have performed a sensitivity analysis of several boundary conditions scenarios, varying inlet/outlet and LA wall movement configurations, using patient-specific imaging data of six LAAO patients (three of them with DRT at follow-up). Mesh and cardiac cycle convergence were also analysed. The boundary conditions scenario that better predicted DRT cases had echocardiography-based velocities at the mitral valve outlet, a generic pressure wave from an AF patient at the pulmonary vein inlets, and a dynamic mesh approach for LA wall deformation, emphasizing the need for patient-specific data for realistic simulations. The obtained promising results need to be further validated with larger cohorts, ideally with ground truth data, but they already offer unique insights on thrombogenic risk in the left atria. Full article
(This article belongs to the Special Issue Development and Applications of Multi-Scale Mathematical Models in Cardiology)
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33 pages, 37135 KiB  
Article
Electro-Mechanical Whole-Heart Digital Twins: A Fully Coupled Multi-Physics Approach
by Tobias Gerach, Steffen Schuler, Jonathan Fröhlich, Laura Lindner, Ekaterina Kovacheva, Robin Moss, Eike Moritz Wülfers, Gunnar Seemann, Christian Wieners and Axel Loewe
Mathematics 2021, 9(11), 1247; https://0-doi-org.brum.beds.ac.uk/10.3390/math9111247 - 29 May 2021
Cited by 50 | Viewed by 7257
Abstract
Mathematical models of the human heart are evolving to become a cornerstone of precision medicine and support clinical decision making by providing a powerful tool to understand the mechanisms underlying pathophysiological conditions. In this study, we present a detailed mathematical description of a [...] Read more.
Mathematical models of the human heart are evolving to become a cornerstone of precision medicine and support clinical decision making by providing a powerful tool to understand the mechanisms underlying pathophysiological conditions. In this study, we present a detailed mathematical description of a fully coupled multi-scale model of the human heart, including electrophysiology, mechanics, and a closed-loop model of circulation. State-of-the-art models based on human physiology are used to describe membrane kinetics, excitation-contraction coupling and active tension generation in the atria and the ventricles. Furthermore, we highlight ways to adapt this framework to patient specific measurements to build digital twins. The validity of the model is demonstrated through simulations on a personalized whole heart geometry based on magnetic resonance imaging data of a healthy volunteer. Additionally, the fully coupled model was employed to evaluate the effects of a typical atrial ablation scar on the cardiovascular system. With this work, we provide an adaptable multi-scale model that allows a comprehensive personalization from ion channels to the organ level enabling digital twin modeling. Full article
(This article belongs to the Special Issue Development and Applications of Multi-Scale Mathematical Models in Cardiology)
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Review

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17 pages, 3018 KiB  
Review
Multiscale Modelling of β-Adrenergic Stimulation in Cardiac Electromechanical Function
by Ruben Doste and Alfonso Bueno-Orovio
Mathematics 2021, 9(15), 1785; https://0-doi-org.brum.beds.ac.uk/10.3390/math9151785 - 28 Jul 2021
Cited by 7 | Viewed by 4704
Abstract
β-adrenergic receptor stimulation (β-ARS) is a physiological mechanism that regulates cardiovascular function under stress conditions or physical exercise. Triggered during the so-called “fight-or-flight” response, the activation of the β-adrenergic receptors located on the cardiomyocyte membrane initiates a phosphorylation cascade of multiple ion channel [...] Read more.
β-adrenergic receptor stimulation (β-ARS) is a physiological mechanism that regulates cardiovascular function under stress conditions or physical exercise. Triggered during the so-called “fight-or-flight” response, the activation of the β-adrenergic receptors located on the cardiomyocyte membrane initiates a phosphorylation cascade of multiple ion channel targets that regulate both cellular excitability and recovery and of different proteins involved in intracellular calcium handling. As a result, β-ARS impacts both the electrophysiological and the mechanical response of the cardiomyocyte. β-ARS also plays a crucial role in several cardiac pathologies, greatly modifying cardiac output and potentially causing arrhythmogenic events. Mathematical patient-specific models are nowadays envisioned as an important tool for the personalised study of cardiac disease, the design of tailored treatments, or to inform risk assessment. Despite that, only a reduced number of computational studies of heart disease have incorporated β-ARS modelling. In this review, we describe the main existing multiscale frameworks to equip cellular models of cardiac electrophysiology with a β-ARS response. We also outline various applications of these multiscale frameworks in the study of cardiac pathology. We end with a discussion of the main current limitations and the future steps that need to be taken to adapt these models to a clinical environment and to incorporate them in organ-level simulations. Full article
(This article belongs to the Special Issue Development and Applications of Multi-Scale Mathematical Models in Cardiology)
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Other

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21 pages, 7098 KiB  
Protocol
A Comparison of Regional Classification Strategies Implemented for the Population Based Approach to Modelling Atrial Fibrillation
by Jordan Elliott, Maria Kristina Belen, Luca Mainardi and Josè Felix Rodriguez Matas
Mathematics 2021, 9(14), 1686; https://0-doi-org.brum.beds.ac.uk/10.3390/math9141686 - 17 Jul 2021
Cited by 4 | Viewed by 1623
Abstract
(1) Background: in silico models are increasingly relied upon to study the mechanisms of atrial fibrillation. Due to the complexity associated with atrial models, cellular variability is often ignored. Recent studies have shown that cellular variability may have a larger impact on electrophysiological [...] Read more.
(1) Background: in silico models are increasingly relied upon to study the mechanisms of atrial fibrillation. Due to the complexity associated with atrial models, cellular variability is often ignored. Recent studies have shown that cellular variability may have a larger impact on electrophysiological behaviour than previously expected. This paper compares two methods for AF remodelling using regional populations. (2) Methods: using 200,000 action potentials, experimental data was used to calibrate healthy atrial regional populations with two cellular models. AF remodelling was applied by directly adjusting maximum channel conductances. AF remodelling was also applied through adjusting biomarkers. The methods were compared upon replication of experimental data. (3) Results: compared to the percentage method, the biomarker approach resulted in smaller changes. RMP, APD20, APD50, and APD90 were changed in the percentage method by up to 11%, 500%, 50%, and 60%, respectively. In the biomarker approach, RMP, APD20, APD50, and APD90 were changed by up to 4.5%, 132%, 50%, and 35%, respectively. (4) Conclusion: applying AF remodelling through biomarker-based clustering resulted in channel conductance changes that were consistent with experimental data, while maintaining the highly non-linear relationships between channel conductances and biomarkers. Directly changing conductances in the healthy regional populations impacted the non-linear relationships and resulted in non-physiological APD20 and APD50 values. Full article
(This article belongs to the Special Issue Development and Applications of Multi-Scale Mathematical Models in Cardiology)
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