Bridging the gap between models based on ordinary, delayed, and fractional differentials equations through integral kernels.

Evolution equations with convolution-type integral operators have a history of study, yet a gap exists in the literature regarding the link between certain convolution kernels and new models, including delayed and fractional differential equations. We demonstrate, starting from the logistic model structure, that classical, delayed, and fractional models are special cases of a framework using a gamma Mittag-Leffler memory kernel. We discuss and classify different types of this general kernel, analyze the asymptotic behavior of the general model, and provide numerical simulations. A detailed classification of the memory kernels is presented through parameter analysis. The fractional models we constructed possess distinctive features as they maintain dimensional balance and explicitly relate fractional orders to past data points. Additionally, we illustrate how our models can reproduce the dynamics of COVID-19 infections in Australia, Brazil, and Peru. Our research expands mathematical modeling by presenting a unified framework that facilitates the incorporation of historical data through the utilization of integro-differential equations, fractional or delayed differential equations, as well as classical systems of ordinary differential equations.

Digital twinning of the human ventricular activation sequence to Clinical 12-lead ECGs and magnetic resonance imaging using realistic Purkinje networks for in silico clinical trials.

Cardiac in silico clinical trials can virtually assess the safety and efficacy of therapies using human-based modelling and simulation. These technologies can provide mechanistic explanations for clinically observed pathological behaviour. Designing virtual cohorts for in silico trials requires exploiting clinical data to capture the physiological variability in the human population. The clinical characterisation of ventricular activation and the Purkinje network is challenging, especially non-invasively. Our study aims to present a novel digital twinning pipeline that can efficiently generate and integrate Purkinje networks into human multiscale biventricular models based on subject-specific clinical 12-lead electrocardiogram and magnetic resonance recordings. Essential novel features of the pipeline are the human-based Purkinje network generation method, personalisation considering ECG R wave progression as well as QRS morphology, and translation from reduced-order Eikonal models to equivalent biophysically-detailed monodomain ones. We demonstrate ECG simulations in line with clinical data with clinical image-based multiscale models with Purkinje in four control subjects and two hypertrophic cardiomyopathy patients (simulated and clinical QRS complexes with Pearson's correlation coefficients > 0.7). Our methods also considered possible differences in the density of Purkinje myocardial junctions in the Eikonal-based inference as regional conduction velocities. These differences translated into regional coupling effects between Purkinje and myocardial models in the monodomain formulation. In summary, we demonstrate a digital twin pipeline enabling simulations yielding clinically consistent ECGs with clinical CMR image-based biventricular multiscale models, including personalised Purkinje in healthy and cardiac disease conditions.

Graph-based homogenisation for modelling cardiac fibrosis.

Fibrosis, the excess of extracellular matrix, can affect, and even block, propagation of action potential in cardiac tissue. This can result in deleterious effects on heart function, but the nature and severity of these effects depend strongly on the localisation of fibrosis and its by-products in cardiac tissue, such as collagen scar formation. Computer simulation is an important means of understanding the complex effects of fibrosis on activation patterns in the heart, but concerns of computational cost place restrictions on the spatial resolution of these simulations. In this work, we present a novel numerical homogenisation technique that uses both Eikonal and graph approaches to allow fine-scale heterogeneities in conductivity to be incorporated into a coarser mesh. Homogenisation achieves this by deriving effective conductivity tensors so that a coarser mesh can then be used for numerical simulation. By taking a graph-based approach, our homogenisation technique functions naturally on irregular grids and does not rely upon any assumptions of periodicity, even implicitly. We present results of action potential propagation through fibrotic tissue in two dimensions that show the graph-based homogenisation technique is an accurate and effective way to capture fine-scale domain information on coarser meshes in the context of sharp-fronted travelling waves of activation. As test problems, we consider excitation propagation in tissue with diffuse fibrosis and through a tunnel-like structure designed to test homogenisation, interaction of an excitation wave with a scar region, and functional re-entry.

Characterisation of Omicron Variant during COVID-19 Pandemic and the Impact of Vaccination, Transmission Rate, Mortality, and Reinfection in South Africa, Germany, and Brazil.

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.

A Validated Mathematical Model of the Cytokine Release Syndrome in Severe COVID-19.

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.

Characterization of the COVID-19 pandemic and the impact of uncertainties, mitigation strategies, and underreporting of cases in South Korea, Italy, and Brazil.

By April 7th, 2020, the Coronavirus disease 2019 (COVID-19) has infected one and a half million people worldwide, accounting for over 80 thousand of deaths in 209 countries and territories around the world. The new and fast dynamics of the pandemic are challenging the health systems of different countries. In the absence of vaccines or effective treatments, mitigation policies, such as social isolation and lock-down of cities, have been adopted, but the results vary among different countries. Some countries were able to control the disease at the moment, as is the case of South Korea. Others, like Italy, are now experiencing the peak of the pandemic. Finally, countries with emerging economies and social issues, like Brazil, are in the initial phase of the pandemic. In this work, we use mathematical models with time-dependent coefficients, techniques of inverse and forward uncertainty quantification, and sensitivity analysis to characterize essential aspects of the COVID-19 in the three countries mentioned above. The model parameters estimated for South Korea revealed effective social distancing and isolation policies, border control, and a high number in the percentage of reported cases. In contrast, underreporting of cases was estimated to be very high in Brazil and Italy. In addition, the model estimated a poor isolation policy at the moment in Brazil, with a reduction of contact around 40%, whereas Italy and South Korea estimated numbers for contact reduction are at 75% and 90%, respectively. This characterization of the COVID-19, in these different countries under different scenarios and phases of the pandemic, supports the importance of mitigation policies, such as social distancing. In addition, it raises serious concerns for socially and economically fragile countries, where underreporting poses additional challenges to the management of the COVID-19 pandemic by significantly increasing the uncertainties regarding its dynamics.

A robust multi-objective optimization framework to capture both cellular and intercellular properties in cardiac cellular model tuning: Analyzing different regions of membrane resistance profile in parameter fitting.

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.

Delay differential equation-based models of cardiac tissue: Efficient implementation and effects on spiral-wave dynamics.

Delay differential equations (DDEs) recently have been used in models of cardiac electrophysiology, particularly in studies focusing on electrical alternans, instabilities, and chaos. A number of processes within cardiac cells involve delays, and DDEs can potentially represent mechanisms that result in complex dynamics both at the cellular level and at the tissue level, including cardiac arrhythmias. However, DDE-based formulations introduce new computational challenges due to the need for storing and retrieving past values of variables at each spatial location. Cardiac tissue simulations that use DDEs may require over 28 GB of memory if the history of variables is not managed carefully. This paper addresses both computational and dynamical issues. First, we present new methods for the numerical solution of DDEs in tissue to mitigate the memory requirements associated with the history of variables. The new methods exploit the different time scales of an action potential to dynamically optimize history size. We find that the proposed methods decrease memory usage by up to 95% in cardiac tissue simulations compared to straightforward history-management algorithms. Second, we use the optimized methods to analyze for the first time the dynamics of wave propagation in two-dimensional cardiac tissue for models that include DDEs. In particular, we study the effects of DDEs on spiral-wave dynamics, including wave breakup and chaos, using a canine myocyte model. We find that by introducing delays to the gating variables governing the calcium current, DDEs can induce spiral-wave breakup in 2D cardiac tissue domains.

Discretization-dependent model for weakly connected excitable media.

Pattern formation has been widely observed in extended chemical and biological processes. Although the biochemical systems are highly heterogeneous, homogenized continuum approaches formed by partial differential equations have been employed frequently. Such approaches are usually justified by the difference of scales between the heterogeneities and the characteristic spatial size of the patterns. Under different conditions, for example, under weak coupling, discrete models are more adequate. However, discrete models may be less manageable, for instance, in terms of numerical implementation and mesh generation, than the associated continuum models. Here we study a model to approach discreteness which permits the computer implementation on general unstructured meshes. The model is cast as a partial differential equation but with a parameter that depends not only on heterogeneities sizes, as in the case of quasicontinuum models, but also on the discretization mesh. Therefore, we refer to it as a discretization-dependent model. We validate the approach in a generic excitable media that simulates three different phenomena: the propagation of action membrane potential in cardiac tissue, in myelinated axons of neurons, and concentration waves in chemical microemulsions.

Performance evaluation of GPU parallelization, space-time adaptive algorithms, and their combination for simulating cardiac electrophysiology.

The use of computer models as a tool for the study and understanding of the complex phenomena of cardiac electrophysiology has attained increased importance nowadays. At the same time, the increased complexity of the biophysical processes translates into complex computational and mathematical models. To speed up cardiac simulations and to allow more precise and realistic uses, 2 different techniques have been traditionally exploited: parallel computing and sophisticated numerical methods. In this work, we combine a modern parallel computing technique based on multicore and graphics processing units (GPUs) and a sophisticated numerical method based on a new space-time adaptive algorithm. We evaluate each technique alone and in different combinations: multicore and GPU, multicore and GPU and space adaptivity, multicore and GPU and space adaptivity and time adaptivity. All the techniques and combinations were evaluated under different scenarios: 3D simulations on slabs, 3D simulations on a ventricular mouse mesh, ie, complex geometry, sinus-rhythm, and arrhythmic conditions. Our results suggest that multicore and GPU accelerate the simulations by an approximate factor of 33×, whereas the speedups attained by the space-time adaptive algorithms were approximately 48. Nevertheless, by combining all the techniques, we obtained speedups that ranged between 165 and 498. The tested methods were able to reduce the execution time of a simulation by more than 498× for a complex cellular model in a slab geometry and by 165× in a realistic heart geometry simulating spiral waves. The proposed methods will allow faster and more realistic simulations in a feasible time with no significant loss of accuracy.

Simulation of Ectopic Pacemakers in the Heart: Multiple Ectopic Beats Generated by Reentry inside Fibrotic Regions.

The inclusion of nonconducting media, mimicking cardiac fibrosis, in two models of cardiac tissue produces the formation of ectopic beats. The fraction of nonconducting media in comparison with the fraction of healthy myocytes and the topological distribution of cells determines the probability of ectopic beat generation. First, a detailed subcellular microscopic model that accounts for the microstructure of the cardiac tissue is constructed and employed for the numerical simulation of action potential propagation. Next, an equivalent discrete model is implemented, which permits a faster integration of the equations. This discrete model is a simplified version of the microscopic model that maintains the distribution of connections between cells. Both models produce similar results when describing action potential propagation in homogeneous tissue; however, they slightly differ in the generation of ectopic beats in heterogeneous tissue. Nevertheless, both models present the generation of reentry inside fibrotic tissues. This kind of reentry restricted to microfibrosis regions can result in the formation of ectopic pacemakers, that is, regions that will generate a series of ectopic stimulus at a fast pacing rate. In turn, such activity has been related to trigger fibrillation in the atria and in the ventricles in clinical and animal studies.

Simulations of complex and microscopic models of cardiac electrophysiology powered by multi-GPU platforms.

Key aspects of cardiac electrophysiology, such as slow conduction, conduction block, and saltatory effects have been the research topic of many studies since they are strongly related to cardiac arrhythmia, reentry, fibrillation, or defibrillation. However, to reproduce these phenomena the numerical models need to use subcellular discretization for the solution of the PDEs and nonuniform, heterogeneous tissue electric conductivity. Due to the high computational costs of simulations that reproduce the fine microstructure of cardiac tissue, previous studies have considered tissue experiments of small or moderate sizes and used simple cardiac cell models. In this paper, we develop a cardiac electrophysiology model that captures the microstructure of cardiac tissue by using a very fine spatial discretization (8 µm) and uses a very modern and complex cell model based on Markov chains for the characterization of ion channel's structure and dynamics. To cope with the computational challenges, the model was parallelized using a hybrid approach: cluster computing and GPGPUs (general-purpose computing on graphics processing units). Our parallel implementation of this model using a multi-GPU platform was able to reduce the execution times of the simulations from more than 6 days (on a single processor) to 21 minutes (on a small 8-node cluster equipped with 16 GPUs, i.e., 2 GPUs per node).

Computational modeling of microabscess formation.

Bacterial infections can be of two types: acute or chronic. The chronic bacterial infections are characterized by being a large bacterial infection and/or an infection where the bacteria grows rapidly. In these cases, the immune response is not capable of completely eliminating the infection which may lead to the formation of a pattern known as microabscess (or abscess). The microabscess is characterized by an area comprising fluids, bacteria, immune cells (mainly neutrophils), and many types of dead cells. This distinct pattern of formation can only be numerically reproduced and studied by models that capture the spatiotemporal dynamics of the human immune system (HIS). In this context, our work aims to develop and implement an initial computational model to study the process of microabscess formation during a bacterial infection.

Influence of ischemic core muscle fibers on surface depolarization potentials in superfused cardiac tissue preparations: a simulation study.

Thin-walled cardiac tissue samples superfused with oxygenated solutions are widely used in experimental studies. However, due to decreased oxygen supply and insufficient wash out of waste products in the inner layers of such preparations, electrophysiological functions could be compromised. Although the cascade of events triggered by cutting off perfusion is well known, it remains unclear as to which degree electrophysiological function in viable surface layers is affected by pathological processes occurring in adjacent tissue. Using a 3D numerical bidomain model, we aim to quantify the impact of superfusion-induced heterogeneities occurring in the depth of the tissue on impulse propagation in superficial layers. Simulations demonstrated that both the pattern of activation as well as the distribution of extracellular potentials close to the surface remain essentially unchanged. This was true also for the electrophysiological properties of cells in the surface layer, where most relevant depolarization parameters varied by less than 5.5 %. The main observed effect on the surface was related to action potential duration that shortened noticeably by 53 % as hypoxia deteriorated. Despite the known limitations of such experimental methods, we conclude that superfusion is adequate for studying impulse propagation and depolarization whereas repolarization studies should consider the influence of pathological processes taking place at the core of tissue sample.

The development of a new computational model for the electromechanics of the human ventricular myocyte.

In this work we present a new electromechanical cardiac myocyte model tailored to reproduce the electrical and force generating activities of human ventricular myocytes. The model was created by coupling two existing models: the ten Tusscher electrophysiology model and the Rice myofilament mechanics model. The parameters of the new model were adjusted in order to replicate the available experimental data for human myocytes. The main challenges in this work were the strong feedbacks between the models, the high non-linearity of the models and mainly the lack of human data to make the adjustments.

A 2D-computer model of atrial tissue based on histographs describes the electro-anatomical impact of microstructure on endocardiac potentials and electric near-fields.

In experiments with cardiac tissue, local conduction is described by waveform analysis of the derivative of the extracellular potential Φ(e) and by the loop morphology of the near-field strength E (the components of the electric field parallel and very close to the tissue surface). The question arises whether the features of these signals can be used to quantify the degree of fibrosis in the heart. A computer model allows us to study the behavior of electric signals at the endocardium with respect to known configurations of microstructure which can not be detected during the electrophysiological experiments. This work presents a 2D-computer model with sub-cellular resolution of atrial micro-conduction in the rabbit heart. It is based on the monodomain equations and digitized histographs from tissue slices obtained post-experimentum. It could be shown that excitation spread in densely coupled regions produces uniform and anisotropic conduction. In contrast, zones with parallel fibers separated by uncoupling interstitial space or connective tissue may show uniform or complex signals depending on pacing site. These results suggest that the analysis of Φ(e) and E combined with multi-site pacing could be used to characterize the type and the size of fibrosis.

Limitations of the homogenized cardiac Monodomain model for the case of low gap junctional coupling.

The cardiac Monodomain model is a mathematical model extensively used in studies of propagation of bioelectric wavefronts in the heart. To be able to use the model for complex and large cardiac simulations, such as the case of whole heart and 3D simulations, some parameters of the model that are known to physiologically vary in space, such as the intracellular conductivity, are traditionally kept constant at effective values. These effective values can be obtained via a mathematical procedure called homogenization. In this work we revisit the classical homogenized monodomain formulation to evaluate its ability to reproduce the situation of low gap junctional coupling. This situation arises in many pathological conditions such as during ischemia. Our numerical results suggest some limitations of the homogenized cardiac Monodomain model under these conditions in terms of computed conduction velocity and Action Potential waverforms.

Experimental and theoretical ventricular electrograms and their relation to electrophysiological gradients in the adult rat heart.

The electrical activity of adult mouse and rat hearts has been analyzed extensively, often as a prerequisite for genetic engineering studies or for the development of rodent models of human diseases. Some aspects of the initiation and conduction of the cardiac action potential in rodents closely resemble those in large mammals. However, rodents have a much higher heart rate and their ventricular action potential is triangular and very short. As a consequence, an interpretation of the electrocardiogram in the mouse and rat remains difficult and controversial. In this study, optical mapping techniques have been applied to an in vitro left ventricular adult rat preparation to obtain patterns of conduction and action potential duration measurements from the epicardial surface. This information has been combined with previously published mathematical models of the rat ventricular myocyte to develop a bidomain model for action potential propagation and electrogram formation in the rat left ventricle. Important insights into the basis for the repolarization waveform in the ventricular electrogram of the adult rat have been obtained. Notably, our model demonstrated that the biphasic shape of the rat ventricular repolarization wave can be explained in terms of the transmural and apex-to-base gradients in action potential duration that exist in the rat left ventricle.

CellML and associated tools and techniques.

We have, in the last few years, witnessed the development and availability of an ever increasing number of computer models that describe complex biological structures and processes. The multi-scale and multi-physics nature of these models makes their development particularly challenging, not only from a biological or biophysical viewpoint but also from a mathematical and computational perspective. In addition, the issue of sharing and reusing such models has proved to be particularly problematic, with the published models often lacking information that is required to accurately reproduce the published results. The International Union of Physiological Sciences Physiome Project was launched in 1997 with the aim of tackling the aforementioned issues by providing a framework for the modelling of the human body. As part of this initiative, the specifications of the CellML mark-up language were released in 2001. Now, more than 7 years later, the time has come to assess the situation, in particular with regard to the tools and techniques that are now available to the modelling community. Thus, after introducing CellML, we review and discuss existing editors, validators, online repository, code generators and simulation environments, as well as the CellML Application Program Interface. We also address possible future directions including the need for additional mark-up languages.

Algebraic multigrid preconditioner for the cardiac bidomain model.

The bidomain equations are considered to be one of the most complete descriptions of the electrical activity in cardiac tissue, but large scale simulations, as resulting from discretization of an entire heart, remain a computational challenge due to the elliptic portion of the problem, the part associated with solving the extracellular potential. In such cases, the use of iterative solvers and parallel computing environments are mandatory to make parameter studies feasible. The preconditioned conjugate gradient (PCG) method is a standard choice for this problem. Although robust, its efficiency greatly depends on the choice of preconditioner. On structured grids, it has been demonstrated that a geometric multigrid preconditioner performs significantly better than an incomplete LU (ILU) preconditioner. However, unstructured grids are often preferred to better represent organ boundaries and allow for coarser discretization in the bath far from cardiac surfaces. Under these circumstances, algebraic multigrid (AMG) methods are advantageous since they compute coarser levels directly from the system matrix itself, thus avoiding the complexity of explicitly generating coarser, geometric grids. In this paper, the performance of an AMG preconditioner (BoomerAMG) is compared with that of the standard ILU preconditioner and a direct solver. BoomerAMG is used in two different ways, as a preconditioner and as a standalone solver. Two 3-D simulation examples modeling the induction of arrhythmias in rabbit ventricles were used to measure performance in both sequential and parallel simulations. It is shown that the AMG preconditioner is very well suited for the solution of the bidomain equation, being clearly superior to ILU preconditioning in all regards, with speedups by factors in the range 5.9-7.7.