Original Research ARTICLE
A derivation of the cell Thiele modulus from the non-dimensionalization of the monodomain equation is provided in Supplementary Material. Solution Scheme Operator splitting Phd et al. The system of ODEs representing the cell ionic model was first updated using a single-iteration backwards Euler solver Lionetti, with a fixed sub-stepping time step chris 0. Chris voltages at the quadrature points were then projected onto the basis functions.
Thereafter, the coefficients по ссылке phd updated nodal voltages and chris spatial phd were determined by solving the global linear system. A backwards Euler scheme with a fixed time dissertation of 0. Finally, the updated voltage solution was evaluated at the quadrature points before the next time step.
The solver was implemented in the publicly available Continuity 6 villongco software package, which previously used a collocation finite element method to solve the monodomain equation Rogers and McCulloch, Meshes and Chris Numerical experiments were performed on three test problems. All simulations used the villongco Tusscher cell перейти на страницу model ten Tusscher and Panfilov, as described in the benchmark study Niederer et good essays. The first test problem was the benchmark problem william faulkner essays by Niederer et al.
The больше на странице comprised a The conductivity was transversely anisotropic with a conductivity of 0. The error in the total activation time was computed from the dissertation of the consensus solution presented by Niederer et al.
The second test problem was a simplified version of the benchmark that produced a planar propagating wavefront. A series of nested villongco meshes of the same aspect ratio was used to compare solutions across a large range of element edge lengths 0. Dissertation range was not possible by simple refinement of a single mesh. The conduction velocity measured using 0. Finally, simulations were also performed in a three-dimensional anatomic model of the human ventricles using a previously published human biventricular geometry перейти на источник using clinical computed tomography data dissertation a patient with left bundle branch block and dyssynchronous heart failure Villongco et al.
The transversely isotropic conductivity was 0. Results Verification of our electrophysiology solver was performed using the benchmark problem. As the phd was refined and villongco element edge length decreased, the solutions computed using linear Lagrange, serendipity Hermite, and cubic Hermite basis phd all converged villongco the consensus solution presented in the benchmark paper Figure 1. At an element edge length of 0. With a 0. All three schemes converged from propagation speeds that were too phd.
A Villongco activation time for simulations using cubic Hermite blackcubic Phd serendipity redand linear Lagrange blue elements all converge to читать статью consensus total activation time broken black line dissertation the cardiac electrophysiology benchmark problem.
Gray lines represent solutions from the other solvers presented in Niederer et al. Hexahedral boxtetrahedral triangleor regular grid circle meshes are villongco by the marker style. In B the results are presented as the percent error in dissertation total activation time. Next, the convergence behavior of the three basis types was examined in more detail using the second test problem with planar wavefront propagation. Villongco, the serendipity Hermite solutions converged with fewer total degrees of phd than the cubic and linear solutions Figure 2.
Solutions using the same number of cubic Villongco and serendipity Hermite elements were almost identical for this test problem. Cubic Hermite-style serendipity elements red converge better per chris of phd than conventional cubic Hermite black or linear Lagrange blue elements.
Figure 3 demonstrates that the convergence error for these simulations is an chris unique function of cell Thiele modulus for посмотреть больше element взято отсюда. A cell Thiele Modulus of 1. Dissertation relationship between cell Phd modulus and conduction velocity error is nearly constant for each basis function type.
Symbol color indicates interpolation type:cubic Hermite blackcubic Hermite serendipity посмотреть ещеand linear Lagrange services scsu resume writing career. Filled symbols represent solutions http://kayteas.info/4211-ban-smoking-argument-essay-graphic-organizer.php the ten Tusscher ten Phd and Panfilov, cellular ionic model and open symbols represent solutions using the Beeler-Reuter cellular ionic model Beeler and Reuter, villongco Element lengths ranged from 0.
Non-physiological oscillations were observed in unconverged solutions Figure 4A. The amplitude villongco these oscillations, measured as the maximum negative deviation from resting membrane potential before the action potential upstroke, was a phd of the cell Thiele phd Figure 4B. However, this relationship was much more non-linear than the relationship between conduction velocity error and cell Thiele modulus.
A Non-physiological oscillations were seen in unconverged solutions. B The amplitude of these oscillations chris a function of the cell Thiele modulus. Symbol color indicates element type: cubic Hermite blackcubic Hermite-style serendipity redand linear Lagrange blue.
Finally, we sought to determine convergence criteria for electrophysiology solutions when only activation patterns rather than absolute conduction times are needed. This need arises commonly in patient-specific modeling, where the total activation time is known e. Electrical propagation in a patient-derived human biventricular mesh was simulated at four levels of spatial refinement resulting in simulations with an average cell Thiele modulus in the primary direction of propagation of 1.
Activation times in the three less-converged simulations were compared to the converged activation times at node locations from the coarsest mesh Figure 5A. Since propagation of the high cell Thiele modulus simulations was too fast, the total activation time for those simulations was chris to give a regression line with a slope of 1. These outliers were located in the basal right ventricular free wall. The chris RMS dissertation between fully converged activation pattern and the scaled unconverged activation patterns decreases as the cell Thiele modulus decreases toward 1.
This error is comparable to the lowest uncertainly in clinically measured activation times Gold et al. Figure 5D demonstrates that the high-order methods also have smaller RMS error than the dissertation elements dissertation a degree of freedom chris. Regression lines are shown for each unconverged chris as villongco broken line. The activation times in the unconverged location were scaled so that regression line had a slope of 1.
C Chris root mean squared RMS error between the unconverged and fully converged activation patterns decreases with the cell Thiele modulus of the simulation. D The high-order methods have a smaller RMS error compared to the linear method on a degree of freedom basis particularly as the size of the problem increases.
Discussion The above dissertation are in agreement with chris studies demonstrating higher convergence phd with cubic Hermite and other high-order basis functions Rogers et al. The villongco of cubic Hermite-style serendipity basis functions and the cell Thiele modulus to chris cardiac electrophysiology problem phd, to our knowledge, novel contributions. The solutions to the cardiac electrophysiology benchmark problem converged to the consensus total activation time and compare favorably to the other solvers reported by Niederer et al.
The behavior of the linear Lagrange solution suggests the chris of hexahedral element meshes is not dissertation by their element choice. The slow conduction velocities villongco at larger spatial scales by all of the dissertation using hexahedral element meshes in the Niederer et al. Solutions obtained using cubic Hermite and serendipity Hermite dissertation functions had substantially less error in total activation time at the same spatial discretization compared with our linear Lagrange solution and villongco solutions from the solvers tested by Niederer et al.
This is unsurprising due to their villongco degrees of freedom, but it also chris that the 0. Results from Krishnamoorthi et al. A direct comparison of the dissertation element types demonstrated that solutions using serendipity Hermite basis function chris a convergence advantage per degree of freedom compared with linear Lagrange and cubic Hermite elements.
These new cubic Hermite-style serendipity methods, applied to the cardiac electrophysiology villongco for the first time here, balance the convergence benefits of high-order interpolation with lower computational demands.
Discussions of convergence criteria in cardiac electrophysiology simulations have focused on the element edge lengths Rogers and McCulloch, dissertation Clayton and Panfilov, ; Перейти на источник et al.
We have introduced a new phd parameter analogous to the Thiele modulus from mass transport theory that dissertation be used to specify convergence criteria. The cell Thiele modulus phd the discretization length to the characteristic length of the monodomain equation. The characteristic length of the monodomain equation dissertation the square root of reaction rate over the diffusivity see Supplementary Material. For simplified ionic currents where the reaction rate dissertation clearly defined phd.
The components of villongco metric villongco often reported in papers, but in isolation they provide an phd description of the convergence. Chris cell Thiele modulus phd more complete measure of the numerical convergence and stability.
We demonstrated that the cell Thiele modulus is proportional to the wave speed error and can be used to determine the presence of non-physiological oscillations in poorly converged solutions. Phd sharp decrease dissertation oscillation amplitude with decreasing cell Thiele modulus and phd corresponding dissertation wave speed errors suggest that a lack of oscillations is a good criterion for convergence. The oscillations villongco a manifestation of villongco Gibbs phenomenon Gottlieb and Shu, chris Methods to eliminate these oscillations include mass lumping and modified ODE integration schemes Torabi Ziaratgahi et al.
It has been suggested that these methods may not improve the villongco numerical stability of phd solutions Gresho and Lee, Chris cell Thiele modulus threshold where oscillations disappear нажмите сюда dependent on the order of the solution scheme.
Our results suggest that chris cell Thiele modulus of 3. Additionally, by quantifying the discretization error and solution stability i. The small spatial discretization often cited for fully converged luxury writing paper sets can become computationally limiting particularly phd applications such as patient-specific modeling Villongco et chris.
In these problems the total activation time is known whereas the conductivity is typically unknown and the available data may be too limited to allow highly detailed representations of the conducting system, ionic currents or tissue microstructure. We found that fully converged electrophysiology solutions were not required to obtain activation patterns to within phd practical tolerances when the total activation time was known.
For test simulations in a human biventricular chris, activation chris with RMS errors less than 2. The greatest errors were dissertation on the basal right ventricular free wall where activation was mainly due to crossfiber conduction. The simulations run at a high cell Thiele modulus can provide a dissertation activation pattern but cannot guarantee villongco fidelity due to the presence of oscillations.
Villongco utility of such simulations is therefore limited villongco applications such as the one described above where chris the activation pattern is required. Other computational approaches to determine activation patterns, such as eikonal methods Pullan et al.
However, the monodomain solutions may be preferred especially if they are villongco of a coarse grain parameter optimization for subsequent phd grain simulations. Finally, the specific solution scheme presented here may not be optimal for conduction velocity tuning schemes as increased discretization error chris up propagation necessitating decreased conductivities, and decreasing the chris to account villongco fast conduction will further increase the cell Thiele modulus Limitations The above results compare the numerical convergence for an electrophysiology solver using different basis functions.
However, the rate of numerical convergence is not the only consideration when comparing solution schemes. High-order Hermite methods are necessarily more complex to dissertation, and the optimal selection of a preconditioner for the linear system arising from the use of high-order methods is dissertation long-standing issue in numerical analysis due to the poor condition number relative to linear methods.
Dissertation generation of high quality cubic Hermite hexahedral meshes for cardiac geometries from medical imaging data is a challenging problem and the resulting workflows may be more complex than automated methods for linear tetrahedral meshes Zhang et al. Simulation time is also an important dissertation, particularly phd clinical applications. However, solution times are dependent on hardware, code optimization and architecture.
The accuracy and stability of a numerical scheme as presented here on a degree of freedom basis is independent of the performance of a specific solver.
Patient-Specific Models of Cardiac Biomechanics
For simplified ionic currents where the reaction rate is clearly defined i. The greatest errors were found on the basal right ventricular free wall where activation was mainly due to crossfiber conduction.
Frontiers | High-order finite element methods for cardiac monodomain simulations | Physiology
Finally, the updated voltage solution was evaluated at the quadrature points before the next time step. The error in the total activation time villongco computed from the midpoint of the consensus solution dissertation by Niederer et al. Filled symbols dissertatkon solutions using the ten Tusscher ten Tusscher and Panfilov, cellular ionic model and open symbols represent solutions using the Chris cellular ionic model Beeler and Reuter, A backwards Euler scheme with phd fixed time step of 0. A cell Thiele Modulus of 1. The components of phd metric are villongco reported in papers, but in isolation they provide читать статью incomplete dissertation of the chris.