Fig 1 - uploaded by Edward Vigmond
Content may be subject to copyright.
(a) Unrolled left endocardium, (b) unrolled right endocardium, and (c) 3-D His-Purkinje system and endocardial surfaces. The left endocardium is on the left.
Source publication
The sawtooth effect refers to how one end of a cardiac cell is depolarized, while the opposite end is hyperpolarized, upon exposure to an exogenous electric field. Although hypothesized, it has not been observed in tissue. The Purkinje system is a one-dimensional (1-D) cable-like system residing on the endocardial surface of the heart and is the mo...
Citations
... The electrical behaviour of the heart has been extensively studied in various mathematical works [1,5,11,12,17,18]. The Purkinje system, initially identified by physiologist J. Purkinje (1787-1869), plays a significant role in cardiac activity and is represented as a one-dimensional tree-like structure [1,5]. ...
... The electrical behaviour of the heart has been extensively studied in various mathematical works [1,5,11,12,17,18]. The Purkinje system, initially identified by physiologist J. Purkinje (1787-1869), plays a significant role in cardiac activity and is represented as a one-dimensional tree-like structure [1,5]. In a normal heart rhythm, the electrical current originates from the sinus node, travels through the two atria, inducing their contractions, and then converges at the atrioventricular node. ...
... Subsequently, it travels along the His bundle and the Purkinje network to reach the ventricular walls, leading to their contraction. The interaction between the myocardium and the Purkinje system has been explored in various ways [1,5,15]. In [19], the authors introduced a model of the junction between the Purkinje network and the myocardium, presenting a mathematical framework for continuous-level coupling conditions. ...
The objective of this paper is to analyze a coupled problem that describes the propagation of the electric wave in the heart. The problem comprises coupled partial differential equations posed on a three-dimensional domain representing the heart and on a one-dimensional tree representing the Purkinje network. Each system of PDEs is itself coupled to ordinary differential equations that describe the electrical activity at the cellular level. We establish the existence of a unique solution, utilizing a fixed-point approach with a judicious and non-conventional choice of functional spaces and contraction.
2010 Mathematics Subject Classification. Primary 35Q92, Secondary: 35R30.
... Since the heart's physiology involves multiple physics systems, e.g. electrophysiology, (passive and active) mechanics and hemodynamics, an effective integrated computational model is very challenging the eikonal/monodomain model [44], the bidomain/bidomain model [47], the monodomain/bidomain model [48], and the monodomain/monodomain model [49,50]. Here, the first model refers to the one used for the Purkinje network and the second to that applied for the myocardium. ...
In previous work, Zhang et al. (2021) developed an integrated smoothed particle hydrodynamics (SPH) method to simulate the principle aspects of cardiac function, including electrophysiology, passive and active mechanical response of the myocardium. As the inclusion of the Purkinje network in electrocardiology is recognized as fundamental to accurately describing the electrical activation in the right and left ventricles, in this paper, we present a multi-order SPH method to handle the electrical propagation through the Purkinje system and in the myocardium with monodomain/monodomain coupling strategy. We first propose an efficient algorithm for network generation on arbitrarily complex surface by exploiting level-set geometry representation and cell-linked list neighbor search algorithm. Then, a reduced-order SPH method is developed to solve the monodomain equation to characterize the fast electrical activation through the Purkinje network. Finally, a multi-order coupling paradigm is introduced to capture the coupled nature of potential propagation arising from the interaction between the network and the myocardium. A set of numerical examples are studied to assess the computational performance, accuracy and versatility of the proposed method. In particular, numerical study performed in realistic left ventricle demonstrates that the present method features all the physiological issues that characterize a heartbeat simulation, including the initiation of the signal in the Purkinje network and the systolic and diastolic phases. As expected, the results underlie the importance of using physiologically realistic Purkinje network for modeling cardiac function.
... As regards the cardiac modeling with inclusion of the Purkinje network, numerical studies have been mainly focused on the myocardium electrosphysiology with different coupling strategies, e.g. the eikonal/eikonal model [33], the eikonal/monodomain model [44], the bidomain/bidomain model [47], the monodomain/bidomain model [48], and the monodomain/monodomain model [49], [50]. Here, the first model refers to the one used for the Purkinje network and the second to that applied for the myocardium. ...
In previous work, Zhang et al. (2021) [1] developed an integrated smoothed particle hydrodynamics (SPH) method to simulate the principle aspects of cardiac function, including electrophysiology, passive and active mechanical response of the myocardium. As the inclusion of the Purkinje network in electrocardiology is recognized as fundamental to accurately describing the electrical activation in the right and left ventricles, in this paper, we present a multi-order SPH method to handle the electrical propagation through the Purkinje system and in the myocardium with monodomain/monodomain coupling strategy. We first propose an efficient algorithm for network generation on arbitrarily complex surface by exploiting level-set geometry representation and cell-linked list neighbor search algorithm. Then, a reduced-order SPH method is developed to solve the one-dimensional monodomain equation to characterize the fast electrical activation through the Purkinje network. Finally, a multi-order coupling paradigm is introduced to capture the coupled nature of potential propagation arising from the interaction between the network and the myocardium. A set of numerical examples are studied to assess the computational performance, accuracy and versatility of the proposed methods. In particular, numerical study performed in realistic left ventricle demonstrates that the present method features all the physiological issues that characterize a heartbeat simulation, including the initiation of the signal in the Purkinje network and the systolic and diastolic phases. As expected, the results underlie the importance of using physiologically realistic Purkinje network for modeling cardiac functions.
... Previous modeling studies investigating the effect of Purkinje fibers on the cardiac response to defibrillation shocks found that the one-dimensional cable structure of Purkinje fibers makes them more excitable than the bulk of myocardial tissue (Boyle et al., 2010;Vigmond and Clements, 2007). The geometry of Purkinje fibers thus appears to contribute to their higher sensitivity to stimulation compared to ventricular muscle. ...
Magnetic resonance imaging (MRI) employs time-varying magnetic gradient fields, which induce electric fields (E-fields) in the patient that can potentially stimulate the heart. The performance of novel gradient systems is increasingly restricted by regulatory safety limits, thus motivating a deeper understanding of cardiac stimulation (CS) in MRI. This thesis investigates the thresholds and mechanisms underlying CS using an integrative approach of modeling and measurements. First, a numerical modeling framework was developed that combines E-field simulations in computational body models with electrophysiological cardiac fiber models to predict stimulation thresholds and sites in the heart. The CS thresholds predicted for two commercial gradient systems were >10-fold higher than the regulatory limits. Second, cardiac magnetostimulation thresholds were measured in ten healthy pigs using magnetic field pulses created by capacitor discharges into a coil. The average threshold E-field in the porcine heart was 92.9 pm 13.5 V/m. CS thresholds predicted in individualized porcine models derived from MR images reproduced the measurements with deviations of <18%, thus demonstrating the validity of the model for the experimental magnetic field waveform. The numerical and experimental results presented in this thesis inform the derivation of safe operational limits for MRI without unnecessarily restricting gradient performance and thus imaging speed and resolution.
... The bidomain/monodomain electrophysiology model has been widely used to study different components of the cardiac electrical network such as the atrial depolarization also including pathologic atrial fibrillation [36,16] or to model the AV-node depolarization [37,38]. The depolarization in the ventricular myocardium has been investigated in a series of works [17,39,40,41] also including the fast conduction Purkinje network [14,21,42], which is needed to reproduce a realistic ventricular depolarization, especially in the presence of infarction [43] or reentry initiation of arrhythmias [44,45,46]. In these works, the geometry of the Purkinje network is generally obtained by applying a growing algorithm to a one-dimensional (1D) network of fibers, which has to be sufficiently dense in order to correctly activate the 3D myocardium [47,48,49]. ...
In this study we present a novel computational model for unprecedented simulations of the whole cardiac electrophysiology. According to the heterogeneous electrophysiologic properties of the heart, the whole cardiac geometry is decomposed into a set of coupled conductive media having different topology and electrical conductivities: (i) a network of slender bundles comprising a fast conduction atrial network, the AV-node and the ventricular bundles; (ii) the Purkinje network; and (iii) the atrial and ventricular myocardium. The propagation of the action potential in these conductive media is governed by the bidomain/monodomain equations, which are discretized in space using an in-house finite volume method and coupled to three different cellular models, the Courtemanche model [1] for the atrial myocytes, the Stewart model [2] for the Purkinje Network and the ten Tusscher-Panfilov model [3] for the ventricular myocytes. The developed numerical model correctly reproduces the cardiac electrophysiology of the whole human heart in healthy and pathologic conditions and it can be tailored to study and optimize resynchronization therapies or invasive surgical procedures. Importantly, the whole solver is GPU-accelerated using CUDA Fortran providing an unprecedented speedup, thus opening the way for systematic parametric studies and uncertainty quantification analyses.
... As regards the cardiac modeling with inclusion of the Purkinje network, numerical studies have been mainly focused on the myocardium electrosphysiology with different coupling strategies, e.g. the eikonal/eikonal model [30], the eikonal/monodomain model [41], the bidomain/bidomain model [44], the monodomain/bidomain model [45], and the monodomain/monodomain model [46,47]. Here, the first model refers to the one used for the Purkinje network and the second to that applied for the myocardium. ...
In previous work, Zhang et al. (2021) \cite{zhang2021integrative} developed an integrated smoothed particle hydrodynamics (SPH) method to simulate the principle aspects of cardiac function, including electrophysiology, passive and active mechanical response of the myocardium. As the inclusion of the Purkinje network in electrocardiology is recognized as fundamental to accurately describing the electrical activation in the right and left ventricles, in this paper, we present a multi-order SPH method to handle the electrical propagation through the Purkinje system and in the myocardium with monodomain/monodomain coupling strategy. We first propose an efficient algorithm for network generation on arbitrarily complex surface by exploiting level-set geometry representation and cell-linked list neighbor search algorithm. Then, a reduced-order SPH method is developed to solve the one-dimensional monodomain equation to characterize the fast electrical activation through the Purkinje network. Finally, a multi-order coupling paradigm is introduced to capture the coupled nature of potential propagation arising from the interaction between the network and the myocardium. A set of numerical examples are studied to assess the computational performance, accuracy and versatility of the proposed methods. In particular, numerical study performed in realistic left ventricle demonstrates that the present method features all the physiological issues that characterize a heartbeat simulation, including the initiation of the signal in the Purkinje network and the systolic and diastolic phases. As expected, the results underlie the importance of using physiologically realistic Purkinje network for modeling cardiac functions.
... It can be represented as a one-dimensional branching cable system with an increased conduction velocity that couples with the ventricular myocardium at Purkinjemyocardial junctions along the endocardial surface. 7 Due to the inherent difficulties in producing anatomically correct or patient-specific Purkinje trees, most ventricular wholeheart models do not include Purkinje fibers. ...
Ventricular arrhythmias are the primary cause of sudden cardiac death and one of the leading causes of mortality worldwide. Whole-heart computational modeling offers a unique approach for studying ventricular arrhythmias, offering vast potential for developing both a mechanistic understanding of ventricular arrhythmias and clinical applications for treatment. In this review, the fundamentals of whole-heart ventricular modeling and current methods of personalizing models using clinical data are presented. From this foundation, the authors summarize recent advances in whole-heart ventricular arrhythmia modeling. Efforts in gaining mechanistic insights into ventricular arrhythmias are discussed, in addition to other applications of models such as the assessment of novel therapeutics. The review emphasizes the unique benefits of computational modeling that allow for insights that are not obtainable by contemporary experimental or clinical means. Additionally, the clinical impact of modeling is explored, demonstrating how patient care is influenced by the information gained from ventricular arrhythmia models. The authors conclude with future perspectives about the direction of whole-heart ventricular arrhythmia modeling, outlining how advances in neural network methodologies hold the potential to reduce computational expense and permit for efficient whole-heart modeling.
... Bidomain simulations were performed using CARP software [11]. Linear tetrahedral elements with 200 µm edge length were used for myocardium and bath (2.5 million elements, 520k nodes), while cubic Hermite line elements of length 100 µm were implemented for the His fibre [12]. A time step of 25 µs was used. ...
In certain cardiac conduction system pathologies, like bundle branch block, block may be proximal, allowing for electrical stimulation of the more disal His bundle to most effectively restore activation. While selective stimulation of the His bundle is sought, surrounding myocardium may also be excited, resulting in nonselective pacing. The myocardium and His bundle have distinct capture thresholds, but the factors affecting whether His bundle pacing is selective or nonselective remain unelucidated.
Objective:
We investigated the properties which affect the capture thresholds in order to improve selective pacing.
Methods:
We performed biophysically detailed, computer simulations of a His fibre running through a septal wedge preparation to compute capture thresholds under various configurations of electrode polarity and orientation.
Results:
The myocardial capture threshold is close to that of the His bundle. The His fibre needed to intersect with the electrode tip to favor its activation. Inserting the electrode fully within the septum increased the myocardial capture threshold. Reversing polarity, to rely on anode break excitation, also increased the ease of selective pacing.
Conclusion:
Model results were consistent with clinical observations. For selective pacing, the tip needs to be in contact with the His fibre and anodal stimulation is preferable.
Significance:
This study provides insight into helping establish electrode and stimulation parameters for selective His bundle pacing in patients.
... V tiss and V pk are the transmembrane voltages of the ventricular tissue and Purkinje cells, respectively. In Eq. 5, R PMJ is a fixed resistance with a value of 15 MΩ, based on the work presented in [33]. The simulated Purkinje network is generated by the algorithm proposed in [34]. ...
In the calibration process, replicating some cellular properties is the main focus, while the importance of membrane resistance ( Rm ) that is primal in the tissue-level modeling is often underestimated. Previously, we presented a framework in which Rm in addition to action potential (AP) waveform was considered in the cellular model fitting. In this paper, we test the hypothesis that this approach for tuning cellular model parameters improves the accuracy of simulations at the tissue level. In doing so, two different sets of single-cell models are generated via independent realizations of our multi-objective optimization approach. In the first set of calibration (Model I), root-mean-square error ( RMSE ) of AP, and absolute error (AE) of maximum upstroke velocity are included as optimization functions; however, in the second set of calibration (Model II), RMSE of Rm curve in the repolarization phase is also added to the optimization functions. The calibrated cell models are then used in several tissue configurations of physiological relevance. We adopt well-defined evaluation metrics to compare tissue models tuned using Models I and II. In the source-sink mismatch configuration, the average absolute relative error ( ARE ) of the critical transition border, defined as the smallest required window width between source and sink for AP propagation, is less than 4.7% in Model II, while this error is increased to more than 8.9% in Model I. In addition, in Model I, the average ARE of total time for activation of tissue is 3.3-6.3%; however, in Model II, this error is reduced to 0.7-1.6%. In the Purkinje-myocardium configuration, the average of RMSE of activation time map is reduced approximately 75% in Model II. Finally, in the transmural APD heterogeneity configuration, the average ARE s of AP duration (APD) and APD dispersion (i.e., the difference between maximum and minimum of APD) are about 13.2% and 17.4% in Model I and 5.8% and 6.8% in Model II, respectively. Overall, our results demonstrate that consideration of Rm in the single-cell optimization procedure yields a substantial improvement in the accuracy of tissue models.
... The Model had high accuracy and can adjust the boundary conditions, but it was not suitable for the simulation of large Deformation of soft tissue without updating the stiffness matrix of finite element. On the basis of rotation invariance, Nesme [7] discussed the finite element method of soft tissue and QR decomposition of the finite element method, which effectively reduced the calculation time of the finite element method. Chanthasopeephan [8] reduced the dimension of the finite element model without reducing the accuracy of the feedback force. ...
The technique of force and haptic reappearance is an effective method to solve the shortage of haptic presence and improve the medical robots' practicability. Soft tissue models, the core of force-haptic reappearance systems, play a decisive role in its performance. The establishment of realistic soft tissue models can improve the system's authenticity and efficiency and better realize the representation of force and touch in the interaction process.
At present, there exists a contradiction between timeliness and accuracy in soft tissue modeling. This paper combined the finite element method with the mass-spring model. We estimated the mass-spring model's parameters with the finite element method by neglecting the damping coefficient and obtained the relationship between the elastic coefficients. Then, according to the real measurement data of soft tissue in literature and the stress-strain curve obtained from real measurement, the values of a, A, ε, ks σ were determined. Through those methods above, an improved soft tissue model was obtained. Through our comparison experiments, the improved spring particle model has a high degree of data fitting, and the force value under the same displacement is smaller. Moreover, the improved model's force-displacement curve in the large deformation stage is still very close to the measurement curve of the volume, which cannot be achieved by the empirical spring particle model. These comparative experiments show that the improved finite element-based particle spring model can better consider the timeliness and accuracy and simulate the soft tissue more accurately.
