Data-driven Tissue Mechanics with Polyconvex Neural Ordinary Differential Equations.

Data-driven methods are becoming an essential part of computational mechanics due to their advantages over traditional material modeling. Deep neural networks are able to learn complex material response without the constraints of closed-form models. However, data-driven approaches do not a priori satisfy physics-based mathematical requirements such as polyconvexity, a condition needed for the existence of minimizers for boundary value problems in elasticity. In this study, we use a recent class of neural networks, neural ordinary differential equations (N-ODEs), to develop data-driven material models that automatically satisfy polyconvexity of the strain energy. We take advantage of the properties of ordinary differential equations to create monotonic functions that approximate the derivatives of the strain energy with respect to deformation invariants. The monotonicity of the derivatives guarantees the convexity of the energy. The N-ODE material model is able to capture synthetic data generated from closed-form material models, and it outperforms conventional models when tested against experimental data on skin, a highly nonlinear and anisotropic material. We also showcase the use of the N-ODE material model in finite element simulations of reconstructive surgery. The framework is general and can be used to model a large class of materials, especially biological soft tissues. We therefore expect our methodology to further enable data-driven methods in computational mechanics.

Classifying Drugs by their Arrhythmogenic Risk Using Machine Learning.

All medications have adverse effects. Among the most serious of these are cardiac arrhythmias. Current paradigms for drug safety evaluation are costly, lengthy, conservative, and impede efficient drug development. Here, we combine multiscale experiment and simulation, high-performance computing, and machine learning to create a risk estimator to stratify new and existing drugs according to their proarrhythmic potential. We capitalize on recent developments in machine learning and integrate information across 10 orders of magnitude in space and time to provide a holistic picture of the effects of drugs, either individually or in combination with other drugs. We show, both experimentally and computationally, that drug-induced arrhythmias are dominated by the interplay between two currents with opposing effects: the rapid delayed rectifier potassium current and the L-type calcium current. Using Gaussian process classification, we create a classifier that stratifies drugs into safe and arrhythmic domains for any combinations of these two currents. We demonstrate that our classifier correctly identifies the risk categories of 22 common drugs exclusively on the basis of their concentrations at 50% current block. Our new risk assessment tool explains under which conditions blocking the L-type calcium current can delay or even entirely suppress arrhythmogenic events. Using machine learning in drug safety evaluation can provide a more accurate and comprehensive mechanistic assessment of the proarrhythmic potential of new drugs. Our study paves the way toward establishing science-based criteria to accelerate drug development, design safer drugs, and reduce heart rhythm disorders.

Visualizing the invisible: The effect of asymptomatic transmission on the outbreak dynamics of COVID-19.

Understanding the outbreak dynamics of the COVID-19 pandemic has important implications for successful containment and mitigation strategies. Recent studies suggest that the population prevalence of SARS-CoV-2 antibodies, a proxy for the number of asymptomatic cases, could be an order of magnitude larger than expected from the number of reported symptomatic cases. Knowing the precise prevalence and contagiousness of asymptomatic transmission is critical to estimate the overall dimension and pandemic potential of COVID-19. However, at this stage, the effect of the asymptomatic population, its size, and its outbreak dynamics remain largely unknown. Here we use reported symptomatic case data in conjunction with antibody seroprevalence studies, a mathematical epidemiology model, and a Bayesian framework to infer the epidemiological characteristics of COVID-19. Our model computes, in real time, the time-varying contact rate of the outbreak, and projects the temporal evolution and credible intervals of the effective reproduction number and the symptomatic, asymptomatic, and recovered populations. Our study quantifies the sensitivity of the outbreak dynamics of COVID-19 to three parameters: the effective reproduction number, the ratio between the symptomatic and asymptomatic populations, and the infectious periods of both groups. For nine distinct locations, our model estimates the fraction of the population that has been infected and recovered by Jun 15, 2020 to 24.15% (95% CI: 20.48%-28.14%) for Heinsberg (NRW, Germany), 2.40% (95% CI: 2.09%-2.76%) for Ada County (ID, USA), 46.19% (95% CI: 45.81%-46.60%) for New York City (NY, USA), 11.26% (95% CI: 7.21%-16.03%) for Santa Clara County (CA, USA), 3.09% (95% CI: 2.27%-4.03%) for Denmark, 12.35% (95% CI: 10.03%-15.18%) for Geneva Canton (Switzerland), 5.24% (95% CI: 4.84%-5.70%) for the Netherlands, 1.53% (95% CI: 0.76%-2.62%) for Rio Grande do Sul (Brazil), and 5.32% (95% CI: 4.77%-5.93%) for Belgium. Our method traces the initial outbreak date in Santa Clara County back to January 20, 2020 (95% CI: December 29, 2019-February 13, 2020). Our results could significantly change our understanding and management of the COVID-19 pandemic: A large asymptomatic population will make isolation, containment, and tracing of individual cases challenging. Instead, managing community transmission through increasing population awareness, promoting physical distancing, and encouraging behavioral changes could become more relevant.

Benchmarking physics-informed frameworks for data-driven hyperelasticity.

Data-driven methods have changed the way we understand and model materials. However, while providing unmatched flexibility, these methods have limitations such as reduced capacity to extrapolate, overfitting, and violation of physics constraints. Recently, frameworks that automatically satisfy these requirements have been proposed. Here we review, extend, and compare three promising data-driven methods: Constitutive Artificial Neural Networks (CANN), Input Convex Neural Networks (ICNN), and Neural Ordinary Differential Equations (NODE). Our formulation expands the strain energy potentials in terms of sums of convex non-decreasing functions of invariants and linear combinations of these. The expansion of the energy is shared across all three methods and guarantees the automatic satisfaction of objectivity, material symmetries, and polyconvexity, essential within the context of hyperelasticity. To benchmark the methods, we train them against rubber and skin stress-strain data. All three approaches capture the data almost perfectly, without overfitting, and have some capacity to extrapolate. This is in contrast to unconstrained neural networks which fail to make physically meaningful predictions outside the training range. Interestingly, the methods find different energy functions even though the prediction on the stress data is nearly identical. The most notable differences are observed in the second derivatives, which could impact performance of numerical solvers. On the rich data used in these benchmarks, the models show the anticipated trade-off between number of parameters and accuracy. Overall, CANN, ICNN and NODE retain the flexibility and accuracy of other data-driven methods without compromising on the physics. These methods are ideal options to model arbitrary hyperelastic material behavior.

Physics-informed neural networks for parameter estimation in blood flow models.

BACKGROUND: Physics-informed neural networks (PINNs) have emerged as a powerful tool for solving inverse problems, especially in cases where no complete information about the system is known and scatter measurements are available. This is especially useful in hemodynamics since the boundary information is often difficult to model, and high-quality blood flow measurements are generally hard to obtain. METHODS: In this work, we use the PINNs methodology for estimating reduced-order model parameters and the full velocity field from scatter 2D noisy measurements in the aorta. Two different flow regimes, stationary and transient were studied. RESULTS: We show robust and relatively accurate parameter estimations when using the method with simulated data, while the velocity reconstruction accuracy shows dependence on the measurement quality and the flow pattern complexity. Comparison with a Kalman filter approach shows similar results when the number of parameters to be estimated is low to medium. For a higher number of parameters, only PINNs were capable of achieving good results. CONCLUSION: The method opens a door to deep-learning-driven methods in the simulations of complex coupled physical systems.

WarpPINN: Cine-MR image registration with physics-informed neural networks.

The diagnosis of heart failure usually includes a global functional assessment, such as ejection fraction measured by magnetic resonance imaging. However, these metrics have low discriminate power to distinguish different cardiomyopathies, which may not affect the global function of the heart. Quantifying local deformations in the form of cardiac strain can provide helpful information, but it remains a challenge. In this work, we introduce WarpPINN, a physics-informed neural network to perform image registration to obtain local metrics of heart deformation. We apply this method to cine magnetic resonance images to estimate the motion during the cardiac cycle. We inform our neural network of the near-incompressibility of cardiac tissue by penalizing the Jacobian of the deformation field. The loss function has two components: an intensity-based similarity term between the reference and the warped template images, and a regularizer that represents the hyperelastic behavior of the tissue. The architecture of the neural network allows us to easily compute the strain via automatic differentiation to assess cardiac activity. We use Fourier feature mappings to overcome the spectral bias of neural networks, allowing us to capture discontinuities in the strain field. The algorithm is tested on synthetic examples and on a cine SSFP MRI benchmark of 15 healthy volunteers, where it is trained to learn the deformation mapping of each case. We outperform current methodologies in landmark tracking and provide physiological strain estimations in the radial and circumferential directions. WarpPINN provides precise measurements of local cardiac deformations that can be used for a better diagnosis of heart failure and can be used for general image registration tasks. Source code is available at https://github.com/fsahli/WarpPINN.

Fast Characterization of Inducible Regions of Atrial Fibrillation Models With Multi-Fidelity Gaussian Process Classification.

Computational models of atrial fibrillation have successfully been used to predict optimal ablation sites. A critical step to assess the effect of an ablation pattern is to pace the model from different, potentially random, locations to determine whether arrhythmias can be induced in the atria. In this work, we propose to use multi-fidelity Gaussian process classification on Riemannian manifolds to efficiently determine the regions in the atria where arrhythmias are inducible. We build a probabilistic classifier that operates directly on the atrial surface. We take advantage of lower resolution models to explore the atrial surface and combine seamlessly with high-resolution models to identify regions of inducibility. We test our methodology in 9 different cases, with different levels of fibrosis and ablation treatments, totalling 1,800 high resolution and 900 low resolution simulations of atrial fibrillation. When trained with 40 samples, our multi-fidelity classifier that combines low and high resolution models, shows a balanced accuracy that is, on average, 5.7% higher than a nearest neighbor classifier. We hope that this new technique will allow faster and more precise clinical applications of computational models for atrial fibrillation. All data and code accompanying this manuscript will be made publicly available at: https://github.com/fsahli/AtrialMFclass.

Sex Differences in Drug-Induced Arrhythmogenesis.

The electrical activity in the heart varies significantly between men and women and results in a sex-specific response to drugs. Recent evidence suggests that women are more than twice as likely as men to develop drug-induced arrhythmia with potentially fatal consequences. Yet, the sex-specific differences in drug-induced arrhythmogenesis remain poorly understood. Here we integrate multiscale modeling and machine learning to gain mechanistic insight into the sex-specific origin of drug-induced cardiac arrhythmia at differing drug concentrations. To quantify critical drug concentrations in male and female hearts, we identify the most important ion channels that trigger male and female arrhythmogenesis, and create and train a sex-specific multi-fidelity arrhythmogenic risk classifier. Our study reveals that sex differences in ion channel activity, tissue conductivity, and heart dimensions trigger longer QT-intervals in women than in men. We quantify the critical drug concentration for dofetilide, a high risk drug, to be seven times lower for women than for men. Our results emphasize the importance of including sex as an independent biological variable in risk assessment during drug development. Acknowledging and understanding sex differences in drug safety evaluation is critical when developing novel therapeutic treatments on a personalized basis. The general trends of this study have significant implications on the development of safe and efficacious new drugs and the prescription of existing drugs in combination with other drugs.

Outbreak dynamics of COVID-19 in Europe and the effect of travel restrictions.

For the first time in history, on March 17, 2020, the European Union closed all its external borders in an attempt to contain the spreading of the coronavirus 2019, COVID-19. Throughout two past months, governments around the world have implemented massive travel restrictions and border control to mitigate the outbreak of this global pandemic. However, the precise effects of travel restrictions on the outbreak dynamics of COVID-19 remain unknown. Here we combine a global network mobility model with a local epidemiology model to simulate and predict the outbreak dynamics and outbreak control of COVID-19 across Europe. We correlate our mobility model to passenger air travel statistics and calibrate our epidemiology model using the number of reported COVID-19 cases for each country. Our simulations show that mobility networks of air travel can predict the emerging global diffusion pattern of a pandemic at the early stages of the outbreak. Our results suggest that an unconstrained mobility would have significantly accelerated the spreading of COVID-19, especially in Central Europe, Spain, and France. Ultimately, our network epidemiology model can inform political decision making and help identify exit strategies from current travel restrictions and total lockdown.

Outbreak dynamics of COVID-19 in China and the United States.

On March 11, 2020, the World Health Organization declared the coronavirus disease 2019, COVID-19, a global pandemic. In an unprecedented collective effort, massive amounts of data are now being collected worldwide to estimate the immediate and long-term impact of this pandemic on the health system and the global economy. However, the precise timeline of the disease, its transmissibility, and the effect of mitigation strategies remain incompletely understood. Here we integrate a global network model with a local epidemic SEIR model to quantify the outbreak dynamics of COVID-19 in China and the United States. For the outbreak in China, in [Formula: see text] provinces, we found a latent period of 2.56 ± 0.72 days, a contact period of 1.47 ± 0.32 days, and an infectious period of 17.82 ± 2.95 days. We postulate that the latent and infectious periods are disease-specific, whereas the contact period is behavior-specific and can vary between different provinces, states, or countries. For the early stages of the outbreak in the United States, in [Formula: see text] states, we adopted the disease-specific values from China and found a contact period of 3.38 ± 0.69 days. Our network model predicts that-without the massive political mitigation strategies that are in place today-the United States would have faced a basic reproduction number of 5.30 ± 0.95 and a nationwide peak of the outbreak on May 10, 2020 with 3 million infections. Our results demonstrate how mathematical modeling can help estimate outbreak dynamics and provide decision guidelines for successful outbreak control. We anticipate that our model will become a valuable tool to estimate the potential of vaccination and quantify the effect of relaxing political measures including total lockdown, shelter in place, and travel restrictions for low-risk subgroups of the population or for the population as a whole.

Predicting critical drug concentrations and torsadogenic risk using a multiscale exposure-response simulator.

Torsades de pointes is a serious side effect of many drugs that can trigger sudden cardiac death, even in patients with structurally normal hearts. Torsadogenic risk has traditionally been correlated with the blockage of a specific potassium channel and a prolonged recovery period in the electrocardiogram. However, the precise mechanisms by which single channel block translates into heart rhythm disorders remain incompletely understood. Here we establish a multiscale exposure-response simulator that converts block-concentration characteristics from single cell recordings into three-dimensional excitation profiles and electrocardiograms to rapidly assess torsadogenic risk. For the drug dofetilide, we characterize the QT interval and heart rate at different drug concentrations and identify the critical concentration at the onset of torsades de pointes: For dofetilide concentrations of 2x, 3x, and 4x, as multiples of the free plasma concentration Cmax = 2.1 nM, the QT interval increased by +62.0%, +71.2%, and +82.3% compared to baseline, and the heart rate changed by -21.7%, -23.3%, and +88.3%. The last number indicates that, at the critical concentration of 4x, the heart spontaneously developed an episode of a torsades-like arrhythmia. Strikingly, this critical drug concentration is higher than the concentration estimated from early afterdepolarizations in single cells and lower than in one-dimensional cable models. Our results highlight the importance of whole heart modeling and explain, at least in part, why current regulatory paradigms often fail to accurately quantify the pro-arrhythmic potential of a drug. Our exposure-response simulator could provide a more mechanistic assessment of pro-arrhythmic risk and help establish science-based guidelines to reduce rhythm disorders, design safer drugs, and accelerate drug development.

Predicting the cardiac toxicity of drugs using a novel multiscale exposure-response simulator.

A common but serious side effect of many drugs is torsades de pointes, a rhythm disorder that can have fatal consequences. Torsadogenic risk has traditionally been associated with blockage of a specific potassium channel and an increased recovery period in the electrocardiogram. However, the mechanisms that trigger torsades de pointes remain incompletely understood. Here we establish a computational model to explore how drug-induced effects propagate from the single channel, via the single cell, to the whole heart level. Our mechanistic exposure-response simulator translates block-concentration characteristics for arbitrary drugs into three-dimensional excitation profiles and electrocardiogram recordings to rapidly assess torsadogenic risk. For the drug of dofetilide, we show that this risk is highly dose-dependent: at a concentration of 1x, QT prolongation is 55% but the heart maintains its regular sinus rhythm; at 5.7x, QT prolongation is 102% and the heart spontaneously transitions into torsades de points; at 30x, QT prolongation is 132% and the heart adapts a quasi-depolarized state with numerous rapidly flickering local excitations. Our simulations suggest that neither potassium channel blockage nor QT interval prolongation alone trigger torsades de pointes. The underlying mechanism predicted by our model is early afterdepolarization, which translates into pronounced U waves in the electrocardiogram, a signature that is correctly predicted by our model. Beyond the risk assessment of existing drugs, our exposure-response simulator can become a powerful tool to optimize the co-administration of drugs and, ultimately, guide the design of new drugs toward reducing life threatening drug-induced rhythm disorders in the heart.

Predicting drug-induced arrhythmias by multiscale modeling.

Drugs often have undesired side effects. In the heart, they can induce lethal arrhythmias such as torsades de pointes. The risk evaluation of a new compound is costly and can take a long time, which often hinders the development of new drugs. Here, we establish a high-resolution, multiscale computational model to quickly assess the cardiac toxicity of new and existing drugs. The input of the model is the drug-specific current block from single cell electrophysiology; the output is the spatio-temporal activation profile and the associated electrocardiogram. We demonstrate the potential of our model for a low-risk drug, ranolazine, and a high-risk drug, quinidine: For ranolazine, our model predicts a prolonged QT interval of 19.4% compared with baseline and a regular sinus rhythm at 60.15 beats per minute. For quinidine, our model predicts a prolonged QT interval of 78.4% and a spontaneous development of torsades de pointes both in the activation profile and in the electrocardiogram. Our model reveals the mechanisms by which electrophysiological abnormalities propagate across the spatio-temporal scales, from specific channel blockage, via altered single cell action potentials and prolonged QT intervals, to the spontaneous emergence of ventricular tachycardia in the form of torsades de pointes. Our model could have important implications for researchers, regulatory agencies, and pharmaceutical companies on rationalizing safe drug development and reducing the time-to-market of new drugs.

Interpreting Activation Mapping of Atrial Fibrillation: A Hybrid Computational/Physiological Study.

Atrial fibrillation is the most common rhythm disorder of the heart associated with a rapid and irregular beating of the upper chambers. Activation mapping remains the gold standard to diagnose and interpret atrial fibrillation. However, fibrillatory activation maps are highly sensitive to far-field effects, and often disagree with other optical mapping modalities. Here we show that computational modeling can identify spurious non-local components of atrial fibrillation electrograms and improve activation mapping. We motivate our approach with a cohort of patients with potential drivers of persistent atrial fibrillation. In a computational study using a monodomain Maleckar model, we demonstrate that in organized rhythms, electrograms successfully track local activation, whereas in atrial fibrillation, electrograms are sensitive to spiral wave distance and number, spiral tip trajectories, and effects of fibrosis. In a clinical study, we analyzed n = 15 patients with persistent atrial fibrillation that was terminated by limited ablation. In five cases, traditional activation maps revealed a spiral wave at sites of termination; in ten cases, electrogram timings were ambiguous and activation maps showed incomplete reentry. By adjusting electrogram timing through computational modeling, we found rotational activation, which was undetectable with conventional methods. Our results demonstrate that computational modeling can identify non-local deflections to improve activation mapping and explain how and where ablation can terminate persistent atrial fibrillation. Our hybrid computational/physiological approach has the potential to optimize map-guided ablation and improve ablation therapy in atrial fibrillation.

Generating Purkinje networks in the human heart.

The Purkinje network is an integral part of the excitation system in the human heart. Yet, to date, there is no in vivo imaging technique to accurately reconstruct its geometry and structure. Computational modeling of the Purkinje network is increasingly recognized as an alternative strategy to visualize, simulate, and understand the role of the Purkinje system. However, most computational models either have to be generated manually, or fail to smoothly cover the irregular surfaces inside the left and right ventricles. Here we present a new algorithm to reliably create robust Purkinje networks within the human heart. We made the source code of this algorithm freely available online. Using Monte Carlo simulations, we demonstrate that the fractal tree algorithm with our new projection method generates denser and more compact Purkinje networks than previous approaches on irregular surfaces. Under similar conditions, our algorithm generates a network with 1219±61 branches, three times more than a conventional algorithm with 419±107 branches. With a coverage of 11±3mm, the surface density of our new Purkije network is twice as dense as the conventional network with 22±7mm. To demonstrate the importance of a dense Purkinje network in cardiac electrophysiology, we simulated three cases of excitation: with our new Purkinje network, with left-sided Purkinje network, and without Purkinje network. Simulations with our new Purkinje network predicted more realistic activation sequences and activation times than simulations without. Six-lead electrocardiograms of the three case studies agreed with the clinical electrocardiograms under physiological conditions, under pathological conditions of right bundle branch block, and under pathological conditions of trifascicular block. Taken together, our results underpin the importance of the Purkinje network in realistic human heart simulations. Human heart modeling has the potential to support the design of personalized strategies for single- or bi-ventricular pacing, radiofrequency ablation, and cardiac defibrillation with the common goal to restore a normal heart rhythm.