Enhanced optimization-based method for the generation of patient-specific models of Purkinje networks.

Cardiac Purkinje networks are a fundamental part of the conduction system and are known to initiate a variety of cardiac arrhythmias. However, patient-specific modeling of Purkinje networks remains a challenge due to their high morphological complexity. This work presents a novel method based on optimization principles for the generation of Purkinje networks that combines geometric and activation accuracy in branch size, bifurcation angles, and Purkinje-ventricular-junction activation times. Three biventricular meshes with increasing levels of complexity are used to evaluate the performance of our approach. Purkinje-tissue coupled monodomain simulations are executed to evaluate the generated networks in a realistic scenario using the most recent Purkinje/ventricular human cellular models and physiological values for the Purkinje-ventricular-junction characteristic delay. The results demonstrate that the new method can generate patient-specific Purkinje networks with controlled morphological metrics and specified local activation times at the Purkinje-ventricular junctions.

A Poroelastic Approach for Modelling Myocardial Oedema in Acute Myocarditis.

Myocarditis is a general set of mechanisms that manifest themselves into the inflammation of the heart muscle. In 2017, more than 3 million people were affected by this disease worldwide, causing about 47,000 deaths. Many aspects of the origin of this disease are well known, but several important questions regarding the disease remain open. One of them is why some patients develop a significantly localised inflammation while others develop a much more diffuse inflammation, reaching across large portions of the heart. Furthermore, the specific role of the pathogenic agent that causes inflammation as well as the interaction with the immune system in the progression of the disease are still under discussion. Providing answers to these crucial questions can have an important impact on patient treatment. In this scenario, computational methods can aid specialists to understand better the relationships between pathogens and the immune system and elucidate why some patients develop diffuse myocarditis. This paper alters a recently developed model to study the myocardial oedema formation in acute infectious myocarditis. The model describes the finite deformation regime using partial differential equations to represent tissue displacement, fluid pressure, fluid phase, and the concentrations of pathogens and leukocytes. A sensitivity analysis was performed to understand better the influence of the most relevant model parameters on the disease dynamics. The results showed that the poroelastic model could reproduce local and diffuse myocarditis dynamics in simplified and complex geometrical domains.

An ensemble of parameters from a robust Markov-based model reproduces L-type calcium currents from different human cardiac myocytes.

The development of modeling structures at the channel level that can integrate subcellular and cell models and properly reproduce different experimental data is of utmost importance in cardiac electrophysiology. In contrast to gate-based models, Markov Chain models are well suited to promote the integration of the subcellular level of the cardiomyocyte to the whole cell. In this paper, we develop Markov Chain models for the L-type Calcium current that can reproduce the electrophysiology of two established human models for the ventricular and Purkinje cells. In addition, instead of presenting a single set of parameters, we present a collection of set of parameters employing Differential Evolution algorithms that can properly reproduce very different protocol data. We show the importance of using an ensemble of a set of parameter values to obtain proper results when considering a second protocol that suppresses calcium inactivation and mimics a pathological condition. We discuss how model discrepancy, data availability, and parameter identifiability can influence the choice of the size of the collection. In summary, we have modified two cardiac models by proposing new Markov Chain models for the L-type Calcium. We keep the original whole-cell dynamics by reproducing the same characteristic action potential and calcium dynamics, whereas the Markov chain-based description of the L-type Calcium channels allows novel small spatial scale simulations of subcellular processes. Finally, the use of collections of parameters was crucial for addressing model discrepancy, identifiability issues, and avoiding fitting parameters overly precisely, i.e., overfitting.

The Quixotic Task of Forecasting Peaks of COVID-19: Rather Focus on Forward and Backward Projections.

Over the last months, mathematical models have been extensively used to help control the COVID-19 pandemic worldwide. Although extremely useful in many tasks, most models have performed poorly in forecasting the pandemic peaks. We investigate this common pitfall by forecasting four countries' pandemic peak: Austria, Germany, Italy, and South Korea. Far from the peaks, our models can forecast the pandemic dynamics 20 days ahead. Nevertheless, when calibrating our models close to the day of the pandemic peak, all forecasts fail. Uncertainty quantification and sensitivity analysis revealed the main obstacle: the misestimation of the transmission rate. Inverse uncertainty quantification has shown that significant changes in transmission rate commonly precede a peak. These changes are a key factor in forecasting the pandemic peak. Long forecasts of the pandemic peak are therefore undermined by the lack of models that can forecast changes in the transmission rate, i.e., how a particular society behaves, changes of mitigation policies, or how society chooses to respond to them. In addition, our studies revealed that even short forecasts of the pandemic peak are challenging. Backward projections have shown us that the correct estimation of any temporal change in the transmission rate is only possible many days ahead. Our results suggest that the distance between a change in the transmission rate and its correct identification in the curve of active infected cases can be as long as 15 days. This is intrinsic to the phenomenon and how it affects epidemic data: a new case is usually only reported after an incubation period followed by a delay associated with the test. In summary, our results suggest the phenomenon itself challenges the task of forecasting the peak of the COVID-19 pandemic when only epidemic data is available. Nevertheless, we show that exciting results can be obtained when using the same models to project different scenarios of reduced transmission rates. Therefore, our results highlight that mathematical modeling can help control COVID-19 pandemic by backward projections that characterize the phenomena' essential features and forward projections when different scenarios and strategies can be tested and used for decision-making.

Validation of a yellow fever vaccine model using data from primary vaccination in children and adults, re-vaccination and dose-response in adults and studies with immunocompromised individuals.

BACKGROUND: An effective yellow fever (YF) vaccine has been available since 1937. Nevertheless, questions regarding its use remain poorly understood, such as the ideal dose to confer immunity against the disease, the need for a booster dose, the optimal immunisation schedule for immunocompetent, immunosuppressed, and pediatric populations, among other issues. This work aims to demonstrate that computational tools can be used to simulate different scenarios regarding YF vaccination and the immune response of individuals to this vaccine, thus assisting the response of some of these open questions. RESULTS: This work presents the computational results obtained by a mathematical model of the human immune response to vaccination against YF. Five scenarios were simulated: primovaccination in adults and children, booster dose in adult individuals, vaccination of individuals with autoimmune diseases under immunomodulatory therapy, and the immune response to different vaccine doses. Where data were available, the model was able to quantitatively replicate the levels of antibodies obtained experimentally. In addition, for those scenarios where data were not available, it was possible to qualitatively reproduce the immune response behaviours described in the literature. CONCLUSIONS: Our simulations show that the minimum dose to confer immunity against YF is half of the reference dose. The results also suggest that immunological immaturity in children limits the induction and persistence of long-lived plasma cells are related to the antibody decay observed experimentally. Finally, the decay observed in the antibody level after ten years suggests that a booster dose is necessary to keep immunity against YF.

Stabilized hybrid discontinuous Galerkin finite element method for the cardiac monodomain equation.

Numerical methods for solving the cardiac electrophysiology model, which describes the electrical activity in the heart, are proposed. The model problem consists of a nonlinear reaction-diffusion partial differential equation coupled to systems of ordinary differential equations that describes electrochemical reactions in cardiac cells. The proposed methods combine an operator splitting technique for the reaction-diffusion equation with primal hybrid methods for spatial discretization considering continuous or discontinuous approximations for the Lagrange multiplier. A static condensation is adopted to form a reduced global system in terms of the multiplier only. Convergence studies exhibit optimal rates of convergence and numerical experiments show that the proposed schemes can be more efficient than standard numerical techniques commonly used in this context when preconditioned iterative methods are used for the solution of linear systems.

A personalized computational model of edema formation in myocarditis based on long-axis biventricular MRI images.

BACKGROUND: Myocarditis is defined as the inflammation of the myocardium, i.e. the cardiac muscle. Among the reasons that lead to this disease, we may include infections caused by a virus, bacteria, protozoa, fungus, and others. One of the signs of the inflammation is the formation of edema, which may be a consequence of the interaction between interstitial fluid dynamics and immune response. This complex physiological process was mathematically modeled using a nonlinear system of partial differential equations (PDE) based on porous media approach. By combing a model based on Biot's poroelasticity theory with a model for the immune response we developed a new hydro-mechanical model for inflammatory edema. To verify this new computational model, T2 parametric mapping obtained by Magnetic Resonance (MR) imaging was used to identify the region of edema in a patient diagnosed with unspecific myocarditis. RESULTS: A patient-specific geometrical model was created using MRI images from the patient with myocarditis. With this model, edema formation was simulated using the proposed hydro-mechanical mathematical model in a two-dimensional domain. The computer simulations allowed us to correlate spatiotemporal dynamics of representative cells of the immune systems, such as leucocytes and the pathogen, with fluid accumulation and cardiac tissue deformation. CONCLUSIONS: This study demonstrates that the proposed mathematical model is a very promising tool to better understand edema formation in myocarditis. Simulations obtained from a patient-specific model reproduced important aspects related to the formation of cardiac edema, its area, position, and shape, and how these features are related to immune response.

Ectopic beats arise from micro-reentries near infarct regions in simulations of a patient-specific heart model.

Ectopic beats are known to be involved in the initiation of a variety of cardiac arrhythmias. Although their location may vary, ectopic excitations have been found to originate from infarct areas, regions of micro-fibrosis and other heterogeneous tissues. However, the underlying mechanisms that link ectopic foci to heterogeneous tissues have yet to be fully understood. In this work, we investigate the mechanism of micro-reentry that leads to the generation of ectopic beats near infarct areas using a patient-specific heart model. The patient-specific geometrical model of the heart, including scar and peri-infarct zones, is obtained through magnetic resonance imaging (MRI). The infarct region is composed of ischemic myocytes and non-conducting cells (fibrosis, for instance). Electrophysiology is captured using an established cardiac myocyte model of the human ventricle modified to describe ischemia. The simulation results clearly reveal that ectopic beats emerge from micro-reentries that are sustained by the heterogeneous structure of the infarct regions. Because microscopic information about the heterogeneous structure of the infarct regions is not available, Monte-Carlo simulations are used to identify the probabilities of an infarct region to behave as an ectopic focus for different levels of ischemia and different percentages of non-conducting cells. From the proposed model, it is observed that ectopic beats are generated when a percentage of non-conducting cells is near a topological metric known as the percolation threshold. Although the mechanism for micro-reentries was proposed half a century ago to be a source of ectopic beats or premature ventricular contractions during myocardial infarction, the present study is the first to reproduce this mechanism in-silico using patient-specific data.

Killing Many Birds With Two Stones: Hypoxia and Fibrosis Can Generate Ectopic Beats in a Human Ventricular Model.

During cardiac diseases many types of anatomical and functional remodeling of cardiac tissue can occur. In this work, we focus on two conditions: hypoxia and fibrosis, which are part of complex pathological modifications that take place in many cardiac diseases (hypertrophic cardiomyopathy, hypertensive heart disease, and recurrent myocardial infarction) and respiratory diseases (obstructive pulmonary disease, obstructive sleep apnea, and cystic fibrosis). Using computational models of cardiac electrophysiology, we evaluate if the interplay between hypoxia and fibrosis is sufficient to trigger cardiac arrhythmia. We study the mechanisms behind the generation of ectopic beats, an arrhythmic trigger also known as premature ventricular contractions (PVCs), in regions with high hypoxia and fibrosis. First, we modify an electrophysiological model of myocytes of the human left ventricle to include the effects of hypoxia. Second, diffuse fibrosis is modeled by randomly replacing cardiac myocytes by non-excitable and non-conducting cells. The Monte Carlo method is used to evaluate the probability of a region to generate ectopic beats with respect to different levels of hypoxia and fibrosis. In addition, we evaluate the minimum size of three-dimensional slabs needed to sustain reentries for different stimulation protocols. The observed mechanism behind the initiation of ectopic beats is unidirectional block, giving rise to sustained micro-reentries inside the region with diffuse fibrosis and hypoxia. In summary, our results suggest that hypoxia and fibrosis are sufficient for the creation of a focal region in the heart that generates PVCs.

Preconditioned augmented Lagrangian formulation for nearly incompressible cardiac mechanics.

Computational modeling of the heart is a subject of substantial medical and scientific interest, which may contribute to increase the understanding of several phenomena associated with cardiac physiological and pathological states. Modeling the mechanics of the heart have led to considerable insights, but it still represents a complex and a demanding computational problem, especially in a strongly coupled electromechanical setting. Passive cardiac tissue is commonly modeled as hyperelastic and is characterized by quasi-incompressible, orthotropic, and nonlinear material behavior. These factors are known to be very challenging for the numerical solution of the model. The near-incompressibility is known to cause numerical issues such as the well-known locking phenomenon and ill-conditioning of the stiffness matrix. In this work, the augmented Lagrangian method is used to handle the nearly incompressible condition. This approach can potentially improve computational performance by reducing the condition number of the stiffness matrix and thereby improving the convergence of iterative solvers. We also improve the performance of iterative solvers by the use of an algebraic multigrid preconditioner. Numerical results of the augmented Lagrangian method combined with a preconditioned iterative solver for a cardiac mechanics benchmark suite are presented to show its improved performance.

Alternans promotion in cardiac electrophysiology models by delay differential equations.

Cardiac electrical alternans is a state of alternation between long and short action potentials and is frequently associated with harmful cardiac conditions. Different dynamic mechanisms can give rise to alternans; however, many cardiac models based on ordinary differential equations are not able to reproduce this phenomenon. A previous study showed that alternans can be induced by the introduction of delay differential equations (DDEs) in the formulations of the ion channel gating variables of a canine myocyte model. The present work demonstrates that this technique is not model-specific by successfully promoting alternans using DDEs for five cardiac electrophysiology models that describe different types of myocytes, with varying degrees of complexity. By analyzing results across the different models, we observe two potential requirements for alternans promotion via DDEs for ionic gates: (i) the gate must have a significant influence on the action potential duration and (ii) a delay must significantly impair the gate's recovery between consecutive action potentials.

Mathematical modeling based on ordinary differential equations: A promising approach to vaccinology.

New contributions that aim to accelerate the development or to improve the efficacy and safety of vaccines arise from many different areas of research and technology. One of these areas is computational science, which traditionally participates in the initial steps, such as the pre-screening of active substances that have the potential to become a vaccine antigen. In this work, we present another promising way to use computational science in vaccinology: mathematical and computational models of important cell and protein dynamics of the immune system. A system of Ordinary Differential Equations represents different immune system populations, such as B cells and T cells, antigen presenting cells and antibodies. In this way, it is possible to simulate, in silico, the immune response to vaccines under development or under study. Distinct scenarios can be simulated by varying parameters of the mathematical model. As a proof of concept, we developed a model of the immune response to vaccination against the yellow fever. Our simulations have shown consistent results when compared with experimental data available in the literature. The model is generic enough to represent the action of other diseases or vaccines in the human immune system, such as dengue and Zika virus.

On the coupling of two models of the human immune response to an antigen.

The development of mathematical models of the immune response allows a better understanding of the multifaceted mechanisms of the defense system. The main purpose of this work is to present a scheme for coupling distinct models of different scales and aspects of the immune system. As an example, we propose a new model where the local tissue inflammation processes are simulated with partial differential equations (PDEs) whereas a system of ordinary differential equations (ODEs) is used as a model for the systemic response. The simulation of distinct scenarios allows the analysis of the dynamics of various immune cells in the presence of an antigen. Preliminary results of this approach with a sensitivity analysis of the coupled model are shown but further validation is still required.

A finite element approach for modeling micro-structural discontinuities in the heart.

The presence of connective tissue as well as interstitial clefts forms a natural barrier to the electrical propagation in the heart. At a microscopic scale, such uncoupling structures change the pattern of the electrical conduction from uniform towards complex and may play a role in the genesis of cardiac arrhythmias. The anatomical diversity of conduction structures and their topology at a microscopic size scale is overwhelming for experimental techniques. Mathematical models have been often employed to study the behavior of the electrical propagation at a sub-cellular level. However, very fine and computationally expensive meshes are required to capture all microscopic details found in the cardiac tissue. In this work, we present a numerical technique based on the finite element method which allows to reproduce the effects of microscopic conduction barriers caused by the presence of uncoupling structures without actually resolving these structures in a high resolution mesh, thereby reducing the computational costs significantly.

ATX-II effects on the apparent location of M cells in a computational model of a human left ventricular wedge.

INTRODUCTION: The apparent location of the myocytes (M cells) with the longest action potential duration (APD) in a canine left ventricular (LV) wedge have been reported to shift after application of a sea anemone toxin, ATX-II. This toxin slows inactivation of I(Na) and thus prolongs APD. Thus, M cells may exhibit dynamic functional states, rather than being a static, anatomically discrete, myocyte population. In this study, we attempted to further define and understand this phenomenon using a mathematical model of the human ventricular myocyte action potential incorporated into an in silico "wedge" preparation. Our simulations demonstrate that even under conditions of a fixed population and ratio of epicardial, M, and endocardial myocytes, the apparent anatomical position (transmural location) of the myocytes with the longest APD can shift following ATX-II treatment. This arises because the ATX-II effect, modeled as a small increase in the late or persistent Na(+) current, and consequent prolongation of APD significantly changes the electrotonic interactions between ventricular myocytes in this LV wedge preparation. METHODS AND RESULTS: This LV wedge model is based on bidomain equations. It corresponds to a rectangular tissue immersed in a passive and isotropic medium that represents the superfusion bath. In this theoretical work, the three known different and discrete populations of myocytes in the human left ventricle have been included: the epicardial, M, and endocardial cells. The effects of ATX-II on I(Na) were simulated by altering the voltage-dependent steady-state inactivation of the parameters h (fast gate) and j (slow gate). As a result, in these ATX-II simulations a persistent late Na(+) current was generated in all three types of ventricular myocytes. However, the APDs were prolonged in a heterogeneous pattern. Our simulations demonstrate that after the ATX-II effects develop, alterations in transmural electrotonic interactions can produce changes in the transmural location of myocytes with the longest APD. CONCLUSIONS: The combination of intercellular electrotonic interactions, which tend to reduce and smooth out the discrete transmural APD variations, and the heterogeneous effects of ATX-II, which preferentially prolong the APD of M cells, can shift the location of the ventricular myocytes. This shift results in significantly altered transmural patterns of action potential durations, which would be expected to change localized refractory period and excitability. These cellular changes give rise to alterations in the corresponding surface electrograms and may change the overall substrates for conduction and rhythm disturbances.