Personalization of electro-mechanical models of the pressure-overloaded left ventricle: fitting of Windkessel-type afterload models

Uncertainty quantification of the pressure waveform using a Windkessel model

The Windkessel (WK) model is a simplified mathematical model used to represent the systemic arterial circulation. While the WK model is useful for studying blood flow dynamics, it suffers from inaccuracies or uncertainties that should be considered when using it to make physiological predictions. This paper aims to develop an efficient and easy-to-implement uncertainty quantification method based on a local gradient-based formulation to quantify the uncertainty of the pressure waveform resulting from aleatory uncertainties of the WK parameters and flow waveform. The proposed methodology, tested against Monte Carlo simulations, demonstrates good agreement in estimating blood pressure uncertainties due to uncertain Windkessel parameters, but less agreement considering uncertain blood-flow waveforms. To illustrate our methodology's applicability, we assessed the aortic pressure uncertainty generated by Windkessel parameters-sets from an available in silico database representing healthy adults. The results from the proposed formulation align qualitatively with those in the database and in vivo data. Furthermore, we investigated how changes in the uncertainty of the Windkessel parameters affect the uncertainty of systolic, diastolic, and pulse pressures. We found that peripheral resistance uncertainty produces the most significant change in the systolic and diastolic blood pressure uncertainties. On the other hand, compliance uncertainty considerably modifies the pulse pressure standard deviation. The presented expansion-based method is a tool for efficiently propagating the Windkessel parameters' uncertainty to the pressure waveform. The Windkessel model's clinical use depends on the reliability of the pressure in the presence of input uncertainties, which can be efficiently investigated with the proposed methodology. For instance, in wearable technology that uses sensor data and the Windkessel model to estimate systolic and diastolic blood pressures, it is important to check the confidence level in these calculations to ensure that the pressures accurately reflect the patient's cardiovascular condition.

Whole-heart electromechanical simulations using Latent Neural Ordinary Differential Equations

Cardiac digital twins provide a physics and physiology informed framework to deliver personalized medicine. However, high-fidelity multi-scale cardiac models remain a barrier to adoption due to their extensive computational costs. Artificial Intelligence-based methods can make the creation of fast and accurate whole-heart digital twins feasible. We use Latent Neural Ordinary Differential Equations (LNODEs) to learn the pressure-volume dynamics of a heart failure patient. Our surrogate model is trained from 400 simulations while accounting for 43 parameters describing cell-to-organ cardiac electromechanics and cardiovascular hemodynamics. LNODEs provide a compact representation of the 3D-0D model in a latent space by means of an Artificial Neural Network that retains only 3 hidden layers with 13 neurons per layer and allows for numerical simulations of cardiac function on a single processor. We employ LNODEs to perform global sensitivity analysis and parameter estimation with uncertainty quantification in 3 hours of computations, still on a single processor.

The role of computational methods in cardiovascular medicine: a narrative review

Background and Objective

Computational models of the cardiovascular system allow for a detailed and quantitative investigation of both physiological and pathological conditions, thanks to their ability to combine clinical-possibly patient-specific-data with physical knowledge of the processes underlying the heart function. These models have been increasingly employed in clinical practice to understand pathological mechanisms and their progression, design medical devices, support clinicians in improving therapies. Hinging upon a long-year experience in cardiovascular modeling, we have recently constructed a computational multi-physics and multi-scale integrated model of the heart for the investigation of its physiological function, the analysis of pathological conditions, and to support clinicians in both diagnosis and treatment planning. This narrative review aims to systematically discuss the role that such model had in addressing specific clinical questions, and how further impact of computational models on clinical practice are envisaged.

Methods

We developed computational models of the physical processes encompassed by the heart function (electrophysiology, electrical activation, force generation, mechanics, blood flow dynamics, valve dynamics, myocardial perfusion) and of their inherently strong coupling. To solve the equations of such models, we devised advanced numerical methods, implemented in a flexible and highly efficient software library. We also developed computational procedures for clinical data post-processing-like the reconstruction of the heart geometry and motion from diagnostic images-and for their integration into computational models.

Key Content and Findings

Our integrated computational model of the heart function provides non-invasive measures of indicators characterizing the heart function and dysfunctions, and sheds light on its underlying processes and their coupling. Moreover, thanks to the close collaboration with several clinical partners, we addressed specific clinical questions on pathological conditions, such as arrhythmias, ventricular dyssynchrony, hypertrophic cardiomyopathy, degeneration of prosthetic valves, and the way coronavirus disease 2019 (COVID-19) infection may affect the cardiac function. In multiple cases, we were also able to provide quantitative indications for treatment.

Conclusions

Computational models provide a quantitative and detailed tool to support clinicians in patient care, which can enhance the assessment of cardiac diseases, the prediction of the development of pathological conditions, and the planning of treatments and follow-up tests.

A comprehensive mathematical model for cardiac perfusion

The aim of this paper is to introduce a new mathematical model that simulates myocardial blood perfusion that accounts for multiscale and multiphysics features. Our model incorporates cardiac electrophysiology, active and passive mechanics, hemodynamics, valve modeling, and a multicompartment Darcy model of perfusion. We consider a fully coupled electromechanical model of the left heart that provides input for a fully coupled Navier-Stokes-Darcy Model for myocardial perfusion. The fluid dynamics problem is modeled in a left heart geometry that includes large epicardial coronaries, while the multicompartment Darcy model is set in a biventricular myocardium. Using a realistic and detailed cardiac geometry, our simulations demonstrate the biophysical fidelity of our model in describing cardiac perfusion. Specifically, we successfully validate the model reliability by comparing in-silico coronary flow rates and average myocardial blood flow with clinically established values ranges reported in relevant literature. Additionally, we investigate the impact of a regurgitant aortic valve on myocardial perfusion, and our results indicate a reduction in myocardial perfusion due to blood flow taken away by the left ventricle during diastole. To the best of our knowledge, our work represents the first instance where electromechanics, hemodynamics, and perfusion are integrated into a single computational framework.

Mechanoelectric effects in healthy cardiac function and under Left Bundle Branch Block pathology

Mechanoelectric feedback (MEF) in the heart operates through several mechanisms which serve to regulate cardiac function. Stretch activated channels (SACs) in the myocyte membrane open in response to cell lengthening, while tension generation depends on stretch, shortening velocity, and calcium concentration. How all of these mechanisms interact and their effect on cardiac output is still not fully understood. We sought to gauge the acute importance of the different MEF mechanisms on heart function. An electromechanical computer model of a dog heart was constructed, using a biventricular geometry of 500K tetrahedral elements. To describe cellular behavior, we used a detailed ionic model to which a SAC model and an active tension model, dependent on stretch and shortening velocity and with calcium sensitivity, were added. Ventricular inflow and outflow were connected to the CircAdapt model of cardiovascular circulation. Pressure-volume loops and activation times were used for model validation. Simulations showed that SACs did not affect acute mechanical response, although if their trigger level was decreased sufficiently, they could cause premature excitations. The stretch dependence of tension had a modest effect in reducing the maximum stretch, and stroke volume, while shortening velocity had a much bigger effect on both. MEF served to reduce the heterogeneity in stretch while increasing tension heterogeneity. In the context of left bundle branch block, a decreased SAC trigger level could restore cardiac output by reducing the maximal stretch when compared to cardiac resynchronization therapy. MEF is an important aspect of cardiac function and could potentially mitigate activation problems.

Fast and robust parameter estimation with uncertainty quantification for the cardiac function

BACKGROUND AND OBJECTIVES

Parameter estimation and uncertainty quantification are crucial in computational cardiology, as they enable the construction of digital twins that faithfully replicate the behavior of physical patients. Many model parameters regarding cardiac electromechanics and cardiovascular hemodynamics need to be robustly fitted by starting from a few, possibly non-invasive, noisy observations. Moreover, short execution times and a small amount of computational resources are required for the effective clinical translation.

METHODS

In the framework of Bayesian statistics, we combine Maximum a Posteriori estimation and Hamiltonian Monte Carlo to find an approximation of model parameters and their posterior distributions. Fast simulations and minimal memory requirements are achieved by using an accurate and geometry-specific Artificial Neural Network surrogate model for the cardiac function, matrix-free methods, automatic differentiation and automatic vectorization. Furthermore, we account for the surrogate modeling error and measurement error.

RESULTS

We perform three different in silico test cases, ranging from the ventricular function to the entire cardiocirculatory system, involving whole-heart mechanics, arterial and venous hemodynamics. By employing a single central processing unit on a standard laptop, we attain highly accurate estimations for all model parameters in short computational times. Furthermore, we obtain posterior distributions that contain the true values inside the 90% credibility regions.

CONCLUSIONS

Many model parameters regarding the entire cardiovascular system can be fastly and robustly identified with minimal hardware requirements. This can be achieved when a small amount of non-invasive data is available and when high levels of signal-to-noise ratio are present in the quantities of interest. With these features, our approach meets the requirements for clinical exploitation, while being compliant with Green Computing practices.

Uncertainty Quantification in the In Vivo Image-Based Estimation of Local Elastic Properties of Vascular Walls

Introduction: Patient-specific computational models are a powerful tool for planning cardiovascular interventions. However, the in vivo patient-specific mechanical properties of vessels represent a major source of uncertainty. In this study, we investigated the effect of uncertainty in the elastic module (E) on a Fluid–Structure Interaction (FSI) model of a patient-specific aorta. Methods: The image-based χ-method was used to compute the initial E value of the vascular wall. The uncertainty quantification was carried out using the generalized Polynomial Chaos (gPC) expansion technique. The stochastic analysis was based on four deterministic simulations considering four quadrature points. A deviation of about ±20% on the estimation of the E value was assumed. Results: The influence of the uncertain E parameter was evaluated along the cardiac cycle on area and flow variations extracted from five cross-sections of the aortic FSI model. Results of stochastic analysis showed the impact of E in the ascending aorta while an insignificant effect was observed in the descending tract. Conclusions: This study demonstrated the importance of the image-based methodology for inferring E, highlighting the feasibility of retrieving useful additional data and enhancing the reliability of in silico models in clinical practice.

Learning Whole Heart Mesh Generation From Patient Images for Computational Simulations

Patient-specific cardiac modeling combines geometries of the heart derived from medical images and biophysical simulations to predict various aspects of cardiac function. However, generating simulation-suitable models of the heart from patient image data often requires complicated procedures and significant human effort. We present a fast and automated deep-learning method to construct simulation-suitable models of the heart from medical images. The approach constructs meshes from 3D patient images by learning to deform a small set of deformation handles on a whole heart template. For both 3D CT and MR data, this method achieves promising accuracy for whole heart reconstruction, consistently outperforming prior methods in constructing simulation-suitable meshes of the heart. When evaluated on time-series CT data, this method produced more anatomically and temporally consistent geometries than prior methods, and was able to produce geometries that better satisfy modeling requirements for cardiac flow simulations. Our source code and pretrained networks are available at https://github.com/fkong7/HeartDeformNets.

Computational analysis of ventricular mechanics in hypertrophic cardiomyopathy patients

Hypertrophic cardiomyopathy (HCM) is a genetic heart disease that is associated with many pathological features, such as a reduction in global longitudinal strain (GLS), myofiber disarray and hypertrophy. The effects of these features on left ventricle (LV) function are, however, not clear in two phenotypes of HCM, namely, obstructive and non-obstructive. To address this issue, we developed patient-specific computational models of the LV using clinical measurements from 2 female HCM patients and a control subject. Left ventricular mechanics was described using an active stress formulation and myofiber disarray was described using a structural tensor in the constitutive models. Unloaded LV configuration for each subject was first determined from their respective end-diastole LV geometries segmented from the cardiac magnetic resonance images, and an empirical single-beat estimation of the end-diastolic pressure volume relationship. The LV was then connected to a closed-loop circulatory model and calibrated using the clinically measured LV pressure and volume waveforms, peak GLS and blood pressure. Without consideration of myofiber disarray, peak myofiber tension was found to be lowest in the obstructive HCM subject (60 kPa), followed by the non-obstructive subject (242 kPa) and the control subject (375 kPa). With increasing myofiber disarray, we found that peak tension has to increase in the HCM models to match the clinical measurements. In the obstructive HCM patient, however, peak tension was still depressed (cf. normal subject) at the largest degree of myofiber disarray found in the clinic. The computational modeling workflow proposed here can be used in future studies with more HCM patient data.

A personalized 0D-1D model of cardiovascular system for the hemodynamic simulation of enhanced external counterpulsation

BACKGROUND AND OBJECTIVE

Enhanced external counterpulsation (EECP) is a non-invasive treatment modality capable of treating a variety of ischemic diseases. Currently, no effective methods of predicting the patient-specific hemodynamic effects of EECP are available. In this study, a personalized 0D-1D model of the cardiovascular system was developed for hemodynamic simulation to simulate the changes in blood flow in the EECP state and develop the best treatment protocol for each individual.

METHODS

A 0D-1D closed-loop model of the cardiovascular system was developed for hemodynamic simulation, consisting of a 1D wave propagation model for arteries, a 0D model for veins and capillaries, and a one-fiber model for the heart. Additionally, a simulation model coupling EECP with a 1D model was established. Physiological data, including the blood flow in different arteries, were clinically collected from 22 volunteers at rest and in the EECP state. Sensitivity analysis and a simulated annealing algorithm were used to build personalized 0D-1D models using the clinical data in the rest state as optimization objectives. Then, the clinical data on EECP were used to verify the applicability and accuracy of the personalized models.

RESULTS

The simulation results and clinical data were found to be in agreement for all 22 subjects, with waveform similarity coefficients (r) exceeding 90% for most arteries at rest and 80% for most arteries during EECP.

CONCLUSIONS

The 0D-1D closed-loop model and the optimized method can facilitate personalized modeling of the cardiovascular system using the data in the rest state and effectively predict the hemodynamic changes in the EECP state, which is significant for the numerical simulation of personalized hemodynamics. The model can also potentially be used to make decisions regarding patient-specific treatment.

Robust and efficient fixed-point algorithm for the inverse elastostatic problem to identify myocardial passive material parameters and the unloaded reference configuration

Image-based computational models of the heart represent a powerful tool to shed new light on the mechanisms underlying physiological and pathological conditions in cardiac function and to improve diagnosis and therapy planning. However, in order to enable the clinical translation of such models, it is crucial to develop personalized models that are able to reproduce the physiological reality of a given patient. There have been numerous contributions in experimental and computational biomechanics to characterize the passive behavior of the myocardium. However, most of these studies suffer from severe limitations and are not applicable to high-resolution geometries. In this work, we present a novel methodology to perform an automated identification of in vivo properties of passive cardiac biomechanics. The highly-efficient algorithm fits material parameters against the shape of a patient-specific approximation of the end-diastolic pressure-volume relation (EDPVR). Simultaneously, an unloaded reference configuration is generated, where a novel line search strategy to improve convergence and robustness is implemented. Only clinical image data or previously generated meshes at one time point during diastole and one measured data point of the EDPVR are required as an input. The proposed method can be straightforwardly coupled to existing finite element (FE) software packages and is applicable to different constitutive laws and FE formulations. Sensitivity analysis demonstrates that the algorithm is robust with respect to initial input parameters.

A coupling strategy for a first 3D-1D model of the cardiovascular system to study the effects of pulse wave propagation on cardiac function

A key factor governing the mechanical performance of the heart is the bidirectional coupling with the vascular system, where alterations in vascular properties modulate the pulsatile load imposed on the heart. Current models of cardiac electromechanics (EM) use simplified 0D representations of the vascular system when coupling to anatomically accurate 3D EM models is considered. However, these ignore important effects related to pulse wave transmission. Accounting for these effects requires 1D models, but a 3D-1D coupling remains challenging. In this work, we propose a novel, stable strategy to couple a 3D cardiac EM model to a 1D model of blood flow in the largest systemic arteries. For the first time, a personalised coupled 3D-1D model of left ventricle and arterial system is built and used in numerical benchmarks to demonstrate robustness and accuracy of our scheme over a range of time steps. Validation of the coupled model is performed by investigating the coupled system's physiological response to variations in the arterial system affecting pulse wave propagation, comprising aortic stiffening, aortic stenosis or bifurcations causing wave reflections. Our first 3D-1D coupled model is shown to be efficient and robust, with negligible additional computational costs compared to 3D-0D models. We further demonstrate that the calibrated 3D-1D model produces simulated data that match with clinical data under baseline conditions, and that known physiological responses to alterations in vascular resistance and stiffness are correctly replicated. Thus, using our coupled 3D-1D model will be beneficial in modelling studies investigating wave propagation phenomena.

An Integrated Workflow for Building Digital Twins of Cardiac Electromechanics-A Multi-Fidelity Approach for Personalising Active Mechanics

Personalised computer models of cardiac function, referred to as cardiac digital twins, are envisioned to play an important role in clinical precision therapies of cardiovascular diseases. A major obstacle hampering clinical translation involves the significant computational costs involved in the personalisation of biophysically detailed mechanistic models that require the identification of high-dimensional parameter vectors. An important aspect to identify in electromechanics (EM) models are active mechanics parameters that govern cardiac contraction and relaxation. In this study, we present a novel, fully automated, and efficient approach for personalising biophysically detailed active mechanics models using a two-step multi-fidelity solution. In the first step, active mechanical behaviour in a given 3D EM model is represented by a purely phenomenological, low-fidelity model, which is personalised at the organ scale by calibration to clinical cavity pressure data. Then, in the second step, median traces of nodal cellular active stress, intracellular calcium concentration, and fibre stretch are generated and utilised to personalise the desired high-fidelity model at the cellular scale using a 0D model of cardiac EM. Our novel approach was tested on a cohort of seven human left ventricular (LV) EM models, created from patients treated for aortic coarctation (CoA). Goodness of fit, computational cost, and robustness of the algorithm against uncertainty in the clinical data and variations of initial guesses were evaluated. We demonstrate that our multi-fidelity approach facilitates the personalisation of a biophysically detailed active stress model within only a few (2 to 4) expensive 3D organ-scale simulations-a computational effort compatible with clinical model applications.

A computationally efficient physiologically comprehensive 3D-0D closed-loop model of the heart and circulation

Computer models of cardiac electro-mechanics (EM) show promise as an effective means for the quantitative analysis of clinical data and, potentially, for predicting therapeutic responses. To realize such advanced applications methodological key challenges must be addressed. Enhanced computational efficiency and robustness is crucial to facilitate, within tractable time frames, model personalization, the simulation of prolonged observation periods under a broad range of conditions, and physiological completeness encompassing therapy-relevant mechanisms is needed to endow models with predictive capabilities beyond the mere replication of observations. Here, we introduce a universal feature-complete cardiac EM modeling framework that builds on a flexible method for coupling a 3D model of bi-ventricular EM to the physiologically comprehensive 0D CircAdapt model representing atrial mechanics and closed-loop circulation. A detailed mathematical description is given and efficiency, robustness, and accuracy of numerical scheme and solver implementation are evaluated. After parameterization and stabilization of the coupled 3D-0D model to a limit cycle under baseline conditions, the model's ability to replicate physiological behaviors is demonstrated, by simulating the transient response to alterations in loading conditions and contractility, as induced by experimental protocols used for assessing systolic and diastolic ventricular properties. Mechanistic completeness and computational efficiency of this novel model render advanced applications geared towards predicting acute outcomes of EM therapies feasible.

A computationally efficient physiologically comprehensive 3D-0D closed-loop model of the heart and circulation

Computer models of cardiac electro-mechanics (EM) show promise as an effective means for the quantitative analysis of clinical data and, potentially, for predicting therapeutic responses. To realize such advanced applications methodological key challenges must be addressed. Enhanced computational efficiency and robustness is crucial to facilitate, within tractable time frames, model personalization, the simulation of prolonged observation periods under a broad range of conditions, and physiological completeness encompassing therapy-relevant mechanisms is needed to endow models with predictive capabilities beyond the mere replication of observations. Here, we introduce a universal feature-complete cardiac EM modeling framework that builds on a flexible method for coupling a 3D model of bi-ventricular EM to the physiologically comprehensive 0D CircAdapt model representing atrial mechanics and closed-loop circulation. A detailed mathematical description is given and efficiency, robustness, and accuracy of numerical scheme and solver implementation are evaluated. After parameterization and stabilization of the coupled 3D-0D model to a limit cycle under baseline conditions, the model's ability to replicate physiological behaviors is demonstrated, by simulating the transient response to alterations in loading conditions and contractility, as induced by experimental protocols used for assessing systolic and diastolic ventricular properties. Mechanistic completeness and computational efficiency of this novel model render advanced applications geared towards predicting acute outcomes of EM therapies feasible.

Effects of Uncertainty of Outlet Boundary Conditions in a Patient-Specific Case of Aortic Coarctation

Computational Fluid Dynamics (CFD) simulations of blood flow are widely used to compute a variety of hemodynamic indicators such as velocity, time-varying wall shear stress, pressure drop, and energy losses. One of the major advances of this approach is that it is non-invasive. The accuracy of the cardiovascular simulations depends directly on the level of certainty on input parameters due to the modelling assumptions or computational settings. Physiologically suitable boundary conditions at the inlet and outlet of the computational domain are needed to perform a patient-specific CFD analysis. These conditions are often affected by uncertainties, whose impact can be quantified through a stochastic approach. A methodology based on a full propagation of the uncertainty from clinical data to model results is proposed here. It was possible to estimate the confidence associated with model predictions, differently than by deterministic simulations. We evaluated the effect of using three-element Windkessel models as the outflow boundary conditions of a patient-specific aortic coarctation model. A parameter was introduced to calibrate the resistances of the Windkessel model at the outlets. The generalized Polynomial Chaos method was adopted to perform the stochastic analysis, starting from a few deterministic simulations. Our results show that the uncertainty of the input parameter gave a remarkable variability on the volume flow rate waveform at the systolic peak simulating the conditions before the treatment. The same uncertain parameter had a slighter effect on other quantities of interest, such as the pressure gradient. Furthermore, the results highlight that the fine-tuning of Windkessel resistances is not necessary to simulate the post-stenting scenario.

Electro-Mechanical Whole-Heart Digital Twins: A Fully Coupled Multi-Physics Approach

Mathematical models of the human heart are evolving to become a cornerstone of precision medicine and support clinical decision making by providing a powerful tool to understand the mechanisms underlying pathophysiological conditions. In this study, we present a detailed mathematical description of a fully coupled multi-scale model of the human heart, including electrophysiology, mechanics, and a closed-loop model of circulation. State-of-the-art models based on human physiology are used to describe membrane kinetics, excitation-contraction coupling and active tension generation in the atria and the ventricles. Furthermore, we highlight ways to adapt this framework to patient specific measurements to build digital twins. The validity of the model is demonstrated through simulations on a personalized whole heart geometry based on magnetic resonance imaging data of a healthy volunteer. Additionally, the fully coupled model was employed to evaluate the effects of a typical atrial ablation scar on the cardiovascular system. With this work, we provide an adaptable multi-scale model that allows a comprehensive personalization from ion channels to the organ level enabling digital twin modeling.