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Bueno-Orovio–Cherry–Fenton model

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The Bueno-Orovio–Cherry–Fenton model, also simply called Bueno-Orovio model, is a minimal ionic model for human ventricular cells.[1] It belongs to the category of phenomenological models, because of its characteristic of describing the electrophysiological behaviour of cardiac muscle cells without taking into account in a detailed way the underlying physiology and the specific mechanisms occurring inside the cells.[2][3]

This mathematical model reproduces both single cell and important tissue-level properties, accounting for physiological action potential development and conduction velocity estimations.[1] It also provides specific parameters choices, derived from parameter-fitting algorithms of the MATLAB Optimization Toolbox, for the modeling of epicardial, endocardial and myd-myocardial tissues.[1] In this way it is possible to match the action potential morphologies, observed from experimental data, in the three different regions of the human ventricles.[1] The Bueno-Orovio–Cherry–Fenton model is also able to describe reentrant and spiral wave dynamics, which occurs for instance during tachycardia or other types of arrhythmias.[1]

From the mathematical perspective, it consists of a system of four differential equations.[1] One PDE, similar to the monodomain model, for an adimensional version of the transmembrane potential, and three ODEs that define the evolution of the so-called gating variables, i.e. probability density functions whose aim is to model the fraction of open ion channels across a cell membrane.[1][4][2]

Mathematical modeling

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Evolution in time only (i.e. case of a single cardiac cell) of the Bueno-Orovio ionic model variables

The system of four differential equations reads as follows:[1]

where is the spatial domain and is the final time. The initial conditions are , , , .[1]

refers to the Heaviside function centered in . The adimensional transmembrane potential can be rescaled in mV by means of the affine transformation .[1] , and refer to gating variables, where in particular can be also used as an indication of intracellular calcium concentration (given in the adimensional range [0, 1] instead of molar concentration).[5]

and are the fast inward, slow outward and slow inward currents respectively, given by the following expressions:[1]

All the above-mentioned ionic density currents are partially adimensional and are expressed in .[1]

Different parameters sets, as shown in Table 1, can be used to reproduce the action potential development of epicardial, endocardial and mid-myocardial human ventricular cells. There are some constants of the model, which are not located in Table 1, that can be deduced with the following formulas:[1]

where the temporal constants, i.e. are expressed in seconds, whereas and are adimensional.[1]

The diffusion coefficient results in a value of , which comes from experimental tests on human ventricular tissues.[1]

In order to trigger the action potential development in a certain position of the domain , a forcing term , which represents an externally applied density current, is usually added at the right hand side of the PDE and acts for a short time interval only.[5]

Table 1: values of the parameters for different positions of the human heart[1]
Parameter
Unity of measure - - - - - - seconds seconds seconds seconds seconds - - seconds seconds seconds seconds seconds seconds - - seconds seconds - - seconds - -
Epicardium 0 1.55 0.3 0.13 0.006 0.006 60e-3 1150e-3 1.4506e-3 60e-3 15e-3 65 0.03 200e-3 0.11e-3 400e-3 6e-3 30.0181e-3 0.9957e-3 2.0458 0.65 2.7342e-3 16e-3 2.0994 0.9087 1.8875e-3 0.07 0.94
Endocardium 0 1.56 0.3 0.13 0.2 0.006 75e-3 10e-3 1.4506e-3 6e-3 140e-3 200 0.016 280e-3 0.1e-3 470e-3 6e-3 40e-3 1.2e-3 2 0.65 2.7342e-3 2e-3 2.0994 0.9087 2.9013e-3 0.0273 0.78
Myocardium 0 1.61 0.3 0.13 0.1 0.005 80e-3 1.4506e-3 1.4506e-3 70e-3 8e-3 200 0.016 280e-3 0.078e-3 410e-3 7e-3 91e-3 0.8e-3 2.1 0.6 2.7342e-3 4e-3 2.0994 0.9087 3.3849e-3 0.01 0.5

Weak formulation

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Assume that refers to the vector containing all the gating variables, i.e. , and contains the corresponding three right hand sides of the ionic model. The Bueno-Orovio–Cherry–Fenton model can be rewritten in the compact form:[6]

Let and be two generic test functions.[6]

To obtain the weak formulation:[6]

  • multiply by the first equation of the model and by the equations for the evolution of the gating variables. Integrating both members of all the equations in the domain :[6]

Finally the weak formulation reads:

Find and , , such that[6]

Numerical discretization

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There are several methods to discretize in space this system of equations, such as the finite element method (FEM) or isogeometric analysis (IGA).[7][8][5][6]

Time discretization can be performed in several ways as well, such as using a backward differentiation formula (BDF) of order or a Runge–Kutta method (RK).[7][5]

Space discretization with FEM

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Let be a tessellation of the computational domain by means of a certain type of elements (such as tetrahedrons or hexahedra), with representing a chosen measure of the size of a single element . Consider the set of polynomials with degree smaller or equal than over an element . Define as the finite dimensional space, whose dimension is . The set of basis functions of is referred to as .[5]

The semidiscretized formulation of the first equation of the model reads: find projection of the solution on , , such that[5]

with , semidiscretized version of the three gating variables, and is the total ionic density current.[5]

The space discretized version of the first equation can be rewritten as a system of non-linear ODEs by setting and :[5]

where , and .[5]

The non-linear term can be treated in different ways, such as using state variable interpolation (SVI) or ionic currents interpolation (ICI).[9][10] In the framework of SVI, by denoting with and the quadrature nodes and weights of a generic element of the mesh , both and are evaluated at the quadrature nodes:[5]

The equations for the three gating variables, which are ODEs, are directly solved in all the degrees of freedom (DOF) of the tessellation separately, leading to the following semidiscrete form:[5]

Time discretization with BDF (implicit scheme)

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With reference to the time interval , let be the chosen time step, with number of subintervals. A uniform partition in time is finally obtained.[7]

At this stage, the full discretization of the Bueno-Orovio ionic model can be performed both in a monolithic and segregated fashion.[11] With respect to the first methodology (the monolithic one), at time , the full problem is entirely solved in one step in order to get by means of either Newton method or Fixed-point iterations:[11]

where and are extrapolations of transmembrane potential and gating variables at previous timesteps with respect to , considering as many time instants as the order of the BDF scheme. is a coefficient which depends on the BDF order .[11]

If a segregated method is employed, the equation for the evolution in time of the transmembrane potential and the ones of the gating variables are numerically solved separately:[11]

  • Firstly, is calculated, using an extrapolation at previous timesteps for the transmembrane potential at the right hand side:[11]
  • Secondly, is computed, exploiting the value of that has just been calculated:[11]

Another possible segregated scheme would be the one in which is calculated first, and then it is used in the equations for .[11]

See also

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References

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  1. ^ a b c d e f g h i j k l m n o p Bueno-Orovio, A.; Cherry, E.M.; Fenton, F. H. (August 2008). "Minimal model for human ventricular action potentials in tissue". Journal of Theoretical Biology. 253 (3): 544–560. doi:10.1016/j.jtbi.2008.03.029. PMID 18495166.
  2. ^ a b Colli Franzone, P.; Pavarino, L.F.; Scacchi, S. (30 October 2014). Mathematical cardiac electrophysiology. Springer. ISBN 978-3-319-04801-7.
  3. ^ Sundnes, J.; Lines, G.T.; Cai, X.; Nielsen, B.F.; Mardal, K.-A.; Tveito, A. (26 June 2007). Computing the electrical activity in the heart. Springer. ISBN 978-3-540-33437-8.
  4. ^ Keener, J.; Sneyd, J. (27 October 2008). Mathematical physiology (2nd ed.). Springer. ISBN 978-0-387-79387-0.
  5. ^ a b c d e f g h i j k Gerbi, A.; Dede’, L.; Quarteroni, A. (2018). "A monolithic algorithm for the simulation of cardiac electromechanics in the human left ventricle". Mathematics in Engineering. 1 (1): 1–37. doi:10.3934/Mine.2018.1.1. hdl:11311/1066015.
  6. ^ a b c d e f g Pegolotti, L.; Dedè, L.; Quarteroni, A. (January 2019). "Isogeometric Analysis of the electrophysiology in the human heart: Numerical simulation of the bidomain equations on the atria" (PDF). Computer Methods in Applied Mechanics and Engineering. 343: 52–73. Bibcode:2019CMAME.343...52P. doi:10.1016/j.cma.2018.08.032.
  7. ^ a b c Quarteroni, A. (25 April 2014). Numerical models for differential problems (Second ed.). Springer. ISBN 978-88-470-5522-3.
  8. ^ Cottrell, J.; Hughes, T.J.R.; Bazilevs, Y. (15 September 2009). Isogeometric analysis: toward integration of CAD and FEA. Wiley. ISBN 978-0-470-74873-2.
  9. ^ Pathmanathan, P.; Mirams, G.R.; Southern, J.; Whiteley, J.P. (November 2011). "The significant effect of the choice of ionic current integration method in cardiac electro-physiological simulations". International Journal for Numerical Methods in Biomedical Engineering. 27 (11): 1751–1770. doi:10.1002/cnm.1438.
  10. ^ Pathmanathan, P.; Bernabeu, M.O.; Niederer, S.A.; Gavaghan, D.J.; Kay, D. (August 2012). "Computational modelling of cardiac electrophysiology: explanation of the variability of results from different numerical solvers". International Journal for Numerical Methods in Biomedical Engineering. 28 (8): 890–903. doi:10.1002/cnm.2467. PMID 25099569.
  11. ^ a b c d e f g Gerbi, A.; Dede', L.; Quarteroni, A. Numerical approximation of cardiac electro-fluid-mechanical models: coupling strategies for large-scale simulation (PDF) (PhD Thesis). École Polytechnique Fédérale de Lausanne.