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Abstract Human-based modelling and simulation offer an ideal testbed for novel medical therapies to guide experimental and clinical studies. Myocardial infarction (MI) is a common cause of heart failure and mortality, for which novel therapies are urgently needed. Although cell therapy offers promise, electrophysiological heterogeneity raises pro-arrhythmic safety concerns, where underlying complex spatio-temporal dynamics cannot be investigated experimentally. Here, after demonstrating credibility of the modelling and simulation framework, we investigate cell therapy in acute versus chronic MI and the role of cell heterogeneity, scar size and the Purkinje system. Simulations agreed with experimental and clinical recordings from ionic to ECG dynamics in acute and chronic infarction. Following cell delivery, spontaneous beats were facilitated by heterogeneity in cell populations, chronic MI due to tissue depolarisation and slow sinus rhythm. Subsequent re-entrant arrhythmias occurred, in some instances with Purkinje involvement and their susceptibility was enhanced by impaired Purkinje-myocardium coupling, large scars and acute infarction. We conclude that homogeneity in injected ventricular-like cell populations minimises their spontaneous beating, which is enhanced by chronic MI, whereas a healthy Purkinje-myocardium coupling is key to prevent subsequent re-entrant arrhythmias, particularly for large scars.

BackgroundVentricular fibrillation (VF) is the deadliest arrhythmia, often caused by myocardial ischaemia. VF patients require urgent intervention planned quickly and non-invasively. However, the accuracy with which electrocardiographic (ECG) markers reflect the underlying arrhythmic substrate is unknown.MethodsWe analysed how ECG metrics reflect the fibrillatory dynamics of electrical excitation and ischaemic substrate. For this, we developed a human-based computational modelling and simulation framework for the quantification of ECG metrics, namely, frequency, slope, and amplitude spectrum area (AMSA) during VF in acute ischaemia for several electrode configurations. Simulations reproduced experimental and clinical findings in 21 scenarios presenting variability in the location and transmural extent of regional ischaemia, and severity of ischaemia in the remote myocardium secondary to VF.ResultsRegional acute myocardial ischaemia facilitated re-entries, potentially breaking up into VF. Ischaemia in the remote myocardium modulated fibrillation dynamics. Cases presenting a mildly ischaemic remote myocardium yielded sustained VF, enabled by the high proliferation of phase singularities (PS, 11–22) causing remarkably disorganised activation patterns. Conversely, global acute ischaemia induced stable rotors (3–12 PS). Changes in frequency and morphology of the ECG during VF reproduced clinical findings but did not show a direct correlation with the underlying wave dynamics. AMSA allowed the precise stratification of VF according to ischaemic severity in the remote myocardium (healthy: 23.62–24.45 mV Hz; mild ischaemia: 10.58–21.47 mV Hz; moderate ischaemia: 4.82–11.12 mV Hz). Within the context of clinical reference values, apex-anterior and apex-posterior electrode configurations were the most discriminatory in stratifying VF based on the underlying ischaemic substrate.ConclusionThis in silico study provides further insights into non-invasive patient-specific strategies for assessing acute ventricular arrhythmias. The use of reliable ECG markers to characterise VF is critical for developing tailored resuscitation strategies.

Abstract This paper uses recurrence quantification analysis (RQA) combined with entropy measures and organization indices to characterize arrhythmic patterns and dynamics in computer simulations of cardiac tissue. We performed different simulations of cardiac tissues of sizes comparable to the human heart atrium. In these simulations, we observed four classic arrhythmic patterns: a spiral wave anchored to a highly fibrotic region resulting in sustained re-entry, a meandering spiral wave, fibrillation, and a spiral wave anchored to a scar region that breaks up into wavelets away from the main rotor. A detailed analysis revealed that, within the same simulation, maps of RQA metrics could differentiate regions with regular AP propagation from ones with chaotic activity. In particular, the combination of two RQA metrics, the length of the longest diagonal string of recurrence points and the mean length of diagonal lines, was able to identify the location of rotor tips, which are the active elements that maintain spiral waves and fibrillation. By proposing low-dimensional models based on the mean value and spatial correlation of metrics calculated from membrane potential time series, we identify RQA-based metrics that successfully separate the four different types of cardiac arrhythmia into distinct regions of the feature space, and thus might be used for automatic classification, in particular distinguishing between fibrillation driven by self-sustaining chaos and that created by a persistent rotor and wavebreak. We also discuss the practical applicability of such an approach.

Abstract 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 resolução de sistemas lineares desempenha um papel fundamental em softwares de simulações computacionais baseadas em modelos matemáticos para o avanço de pesquisas científicas contemporâneas. Consequentemente, há uma demanda crescente por métodos numéricos e implementações eficientes para enfrentar esse desafio, em particular no contexto da engenharia biomédica onde deseja-se utilizar esses simuladores para criar gêmeos digitais de pacientes e estudar determinadas condições patológicas. Este trabalho tem como objetivo explorar e identificar técnicas eficientes para resolver sistemas lineares relacionados ao problema da biomecânica cardíaca, acelerando assim as simulações relacionadas ao intrincado sistema cardiovascular humano. Para atingir esse objetivo, foram selecionados vários métodos multigrid disponíveis na biblioteca AmgX, que foram testados e analisados em termos do seu desempenho computacional. Como um passo inicial, problemas baseados na equação de Poisson, foram resolvidos considerando geometrias simplificadas e complexas como, por exemplo, um cubo e um ventrículo humano. Esse estudo revelou vantagens distintas associadas a cada método, dependendo da complexidade e do formato dos problemas em questão.

Introduction: Skeletal muscle is responsible for multiple functions for maintaining energy homeostasis and daily activities. Muscle contraction is activated by nerve signals, causing calcium release and interaction with myofibrils. It is important to understand muscle behavior and its impact on medical conditions, like in the presence of some diseases and their treatment, such as cancer, which can affect muscle architecture, leading to deficits in its function. For instance, it is known that radiotherapy and chemotherapy also have effects on healthy tissues, leading to a reduction in the rate of force development and the atrophy of muscle fibers. The main aim is to reproduce the behavior of muscle contraction using a coupled model of force generation and the action potential of the cell membrane, inserting the latency period observed between action potential and force generation in the motor unit.Methods: Mathematical models for calcium dynamics and muscle contraction are described, incorporating the role of calcium ions and rates of reaction. An action potential initiates muscle contraction, as described by the Hodgkin–Huxley model. The numerical method used to solve the equations is the forward Euler method.Results and Discussion: The results show dynamic calcium release and force generation, aligning with previous research results, and the time interval between membrane excitation and force generation was accomplished. Future work should suggest simulating more motor units at the actual scale for the possibility of a comparison with real data collected from both healthy individuals and those who have undergone cancer treatment.

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.

Abstract 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.

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.

Cell-based mathematical models have previously been developed to simulate the immune system in response to pathogens. Mathematical modeling papers which study the human immune response to pathogens have predicted concentrations of a variety of cells, including activated and resting macrophages, plasma cells, and antibodies. This study aims to create a comprehensive mathematical model that can predict cytokine levels in response to a gram-positive bacterium, S. aureus by coupling previous models. To accomplish this, the cytokines Tumor Necrosis Factor Alpha (TNF-α), Interleukin 6 (IL-6), Interleukin 8 (IL-8), and Interleukin 10 (IL-10) are included to quantify the relationship between cytokine release from macrophages and the concentration of the pathogen, S. aureus, ex vivo. Partial differential equations (PDEs) are used to model cellular response and ordinary differential equations (ODEs) are used to model cytokine response, and interactions between both components produce a more robust and more complete systems-level understanding of immune activation. In the coupled cellular and cytokine model outlined in this paper, a low concentration of S. aureus is used to stimulate the measured cellular response and cytokine expression. Results show that our cellular activation and cytokine expression model characterizing septic conditions can predict ex vivo mechanisms in response to gram-negative and gram-positive bacteria. Our simulations provide new insights into how the human immune system responds to infections from different pathogens. Novel applications of these insights help in the development of more powerful tools and protocols in infection biology.

Cardiac fibrosis and other scarring of the heart, arising from conditions ranging from myocardial infarction to ageing, promotes dangerous arrhythmias by blocking the healthy propagation of cardiac excitation. Owing to the complexity of the dynamics of electrical signalling in the heart, however, the connection between different arrangements of blockage and various arrhythmic consequences remains poorly understood. Where a mechanism defies traditional understanding, machine learning can be invaluable for enabling accurate prediction of quantities of interest (measures of arrhythmic risk) in terms of predictor variables (such as the arrangement or pattern of obstructive scarring). In this study, we simulate the propagation of the action potential (AP) in tissue affected by fibrotic changes and hence detect sites that initiate re-entrant activation patterns. By separately considering multiple different stimulus regimes, we directly observe and quantify the sensitivity of re-entry formation to activation sequence in the fibrotic region. Then, by extracting the fibrotic structures around locations that both do and do not initiate re-entries, we use neural networks to determine to what extent re-entry initiation is predictable, and over what spatial scale conduction heterogeneities appear to act to produce this effect. We find that structural information within about 0.5 mm of a given point is sufficient to predict structures that initiate re-entry with more than 90% accuracy.

By June 2021, a new contagious disease, the Coronavirus disease 2019 (COVID-19), has infected more than 172 million people worldwide, causing more than 3.7 million deaths. Many aspects related to the interactions of the disease’s causative agent, SAR2-CoV-2, and the immune response are not well understood: the multiscale interactions among the various components of the human immune system and the pathogen are very complex. Mathematical and computational tools can help researchers to answer these open questions about the disease. In this work, we present a system of fifteen ordinary differential equations that models the immune response to SARS-CoV-2. The model is used to investigate the hypothesis that the SARS-CoV-2 infects immune cells and, for this reason, induces high-level productions of inflammatory cytokines. Simulation results support this hypothesis and further explain why survivors have lower levels of cytokines levels than non-survivors.

Abstract 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.

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.

Several variants of SARS-CoV-2 have been identified in different parts of the world, including Gamma, detected in Brazil, Delta, detected in India, and the recent Omicron variant, detected in South Africa. The emergence of a new variant is a cause of great concern. This work considers an extended version of an SIRD model capable of incorporating the effects of vaccination, time-dependent transmissibility rates, mortality, and even potential reinfections during the pandemic. We use this model to characterise the Omicron wave in Brazil, South Africa, and Germany. During Omicron, the transmissibility increased by five for Brazil and Germany and eight for South Africa, whereas the estimated mortality was reduced by three-fold. We estimated that the reported cases accounted for less than 25% of the actual cases during Omicron. The mortality among the nonvaccinated population in these countries is, on average, three to four times higher than the mortality among the fully vaccinated. Finally, we could only reproduce the observed dynamics after introducing a new parameter that accounts for the percentage of the population that can be reinfected. Reinfection was as high as 40% in South Africa, which has only 29% of its population fully vaccinated and as low as 13% in Brazil, which has over 70% and 80% of its population fully vaccinated and with at least one dose, respectively. The calibrated models were able to estimate essential features of the complex virus and vaccination dynamics and stand as valuable tools for quantifying the impact of protocols and decisions in different populations.

Para modelar o escoamento sanguíneo no sistema arterial, é necessário truncar os pequenos domínios arteriais representando as redes das pequenas artérias e arteríolas. Nestes pontos de truncamento, a impedância de entrada é utilizada como condição de contorno, e são inseridos modelos computacionais de árvores arteriais para representar os distritos truncados. Na literatura, encontram-se relatados os seguintes modelos para a representação dos distritos truncados: resistência constante, Windkessel e modelo de árvore estrurada gerado usando leis fractais. Neste trabalho, propõem-se: (i) gerar modelos de árvores arteriais usando o método CCO (Constrained Constructive Optimization) e, em seguida, (ii) quantificar a impedância de entrada destes modelos. Os resultados mostram que a impedância é afetada com a escolha de parâmetros para construção do modelo.

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.

Mathematical models of cardiac cells have been established to broaden understanding of cardiac function. In the process of developing electrophysiological models for cardiac myocytes, precise parameter tuning is a crucial step. The membrane resistance (Rm) is an essential feature obtained from cardiac myocytes. This feature reflects intercellular coupling and affects important phenomena, such as conduction velocity, and early after-depolarizations, but it is often overlooked during the phase of parameter fitting. Thus, the traditional parameter fitting that only includes action potential (AP) waveform may yield incorrect values for Rm. In this paper, a novel multi-objective parameter fitting formulation is proposed and tested that includes different regions of the Rm profile as additional objective functions for optimization. As Rm depends on the transmembrane voltage (Vm) and exhibits singularities for some specific values of Vm, analyses are conducted to carefully select the regions of interest for the proper characterization of Rm. Non-dominated sorting genetic algorithm II is utilized to solve the proposed multi-objective optimization problem. To verify the efficacy of the proposed problem formulation, case studies and comparisons are carried out using multiple models of human cardiac ventricular cells. Results demonstrate Rm is correctly reproduced by the tuned cell models after considering the curve of Rm obtained from the late phase of repolarization and Rm value calculated in the rest phase as additional objectives. However, relative deterioration of the AP fit is observed, demonstrating trade-off among the objectives. This framework can be useful for a wide range of applications, including the parameters fitting phase of the cardiac cell model development and investigation of normal and pathological scenarios in which reproducing both cellular and intercellular properties are of great importance.

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.

Ventricular tachycardia (VT) secondary to myocardial infarction (MI) remain a major cause of sudden death in adults. Premature ventricular complexes (PVCs), the first initiating beats of a portion of these arrhythmias, arise from triggered activity in the infarct border zone (BZ). At the cellular scale, spontaneous calcium release (SCR) events are a known cause of triggered activity and have been reported in cells that survive MI. At the tissue scale, fibrosis has been shown to play an important role in creating the substrate for VT. However, the interplay between SCR-mediated triggered activity and fibrosis upon VT formation in infarcted hearts has not been fully investigated. Here, we conduct in-silico experiments to assess how macroscopic and microscopic anatomical properties of the BZ can create a substrate for SCR-mediated VT formation. To study this question, we employ a stochastic subcellular-scale model of SCR events and action potential to simulate different cardiac preparations. Within 2D sheet models with idealized infarct scars and BZ we show that the probability of PVCs is higher, 55%, in preparations with thin conducting isthmuses (0.2 mm) transcending the scar. In an anatomically-detailed model of the rabbit ventricles with a realistic representation of intramural scars, we show that the heart's protective source-sink mismatch prevents ectopy. Furthermore, we demonstrate that fibrosis disrupts this antiarrhythmic mechanism making PVCs more likely. PVC probability is highest (≥25%) when fibrosis accounts for 60 and 90% of the BZ in the 2D sheet and the 3D anatomical model, respectively. Above these thresholds, PVC occurrence decreases because of: (1) the reduced number of myocytes in the BZ; (2) conduction block. Block is caused either by disconnection of BZ cells from the myocardium or due to source-sink mismatches at regions of rapid tissue expansion. Moreover, while outward propagation to healthy tissue may fail, PVCs traveling inward through the scar might encounter more favorable loading conditions. These PVCs may exit to the myocardium and reenter back at the region of block. Overall, our findings indicate that thin isthmuses and strands of myocytes interspersed with fibrosis can be arrhythmogenic. Ablation of these microscopic structures may prevent VT formation.

Arrhythmias in cardiac tissue are generally associated with irregular electrical wave propagation in the heart. Cardiac tissue is formed by a discrete cell network, which is often heterogeneous. Recently, it was shown in simulations of two-dimensional (2D) discrete models of cardiac tissue that a wave crossing a fibrotic, heterogeneous region may produce reentry and transient or persistent ectopic activity provided the fraction of conducting connections is just above the percolation threshold. Here, we investigate the occurrence of these phenomena in three-dimensions by simulations of a discrete model representing a thin slab of cardiac tissue. This is motivated (i) by the necessity to study the relevance and properties of the percolation-related mechanism for the emergence of microreentries in three dimensions and (ii) by the fact that atrial tissue is quite thin in comparison with ventricular tissue. Here, we simplify the model by neglecting details of tissue anatomy, e. g. geometries of atria or ventricles and the anisotropy in the conductivity. Hence, our modeling study is confined to the investigation of the effect of the tissue thickness as well as to the comparison of the dynamics of electrical excitation in a 2D layer with the one in a 3D slab. Our results indicate a strong and non-trivial effect of the thickness even for thin tissue slabs on the probability of microreentries and ectopic beat generation. The strong correlation of the occurrence of microreentry with the percolation threshold reported earlier in 2D layers persists in 3D slabs. Finally, a qualitative agreement of 3D simulated electrograms in the fibrotic region with the experimentally observed complex fractional atrial electrograms (CFAE) as well as strong difference between simulated electrograms in 2D and 3D were found for the cases where reentry and ectopic activity were triggered by the micro-fibrotic region.