Phenomenological ionic model for human ventricular cells
The Bueno-Orovio–Cherry–Fenton model, also simply called Bueno-Orovio model, is a minimal ionic model for human ventricular cells. 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.
This mathematical model reproduces both single cell and important tissue-level properties, accounting for physiological action potential development and conduction velocity estimations. 
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. 
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.
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.
From the mathematical perspective, it consists of a system of four differential equations. 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.
Mathematical modeling.
The system of four differential equations reads as follows:
 formula_1
where formula_2 is the spatial domain and formula_3 is the final time. The initial conditions are formula_4, formula_5, formula_6, formula_7.
formula_8 refers to the Heaviside function centered in formula_9. The adimensional transmembrane potential formula_10 can be rescaled in mV by means of the affine transformation formula_11. formula_12, formula_13 and formula_14 refer to gating variables, where in particular formula_14 can be also used as an indication of intracellular calcium formula_16
 formula_17
 formula_18
All the above-mentioned ionic density currents are partially adimensional and are expressed in formula_19.
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:
 formula_20
 formula_21
 formula_22
with formula_23, formula_24 semidiscretized version of the three gating variables, and formula_25 is the total ionic density current.
The space discretized version of the first equation can be rewritten as a system of non-linear ODEs by setting formula_26 and formula_27:
 formula_28
where formula_29, formula_30 and
formula_31.
The non-linear term formula_32 can be treated in different ways, such as using state variable interpolation (SVI) or ionic currents interpolation (ICI).
In the framework of SVI, by denoting with formula_33 and formula_34 the quadrature nodes and weights of a generic element of the mesh formula_35, both formula_36 and formula_37 are evaluated at the quadrature nodes:
 formula_38
The equations for the three gating variables, which are ODEs, are directly solved in all the degrees of freedom (DOF) of the tessellation formula_39 separately, leading to the following semidiscrete form:
 formula_40
Time discretization with BDF (implicit scheme).
With reference to the time interval formula_41, let formula_42 be the chosen time step, with formula_43 number of subintervals. A uniform partition in time formula_44 is finally obtained.
At this stage, the full discretization of the Bueno-Orovio ionic model can be performed both in a monolithic and segregated fashion.
With respect to the first methodology (the monolithic one), at time formula_45, the full problem is entirely solved in one step in order to get formula_46 by means of either Newton method or Fixed-point iterations:
 formula_47
where formula_48 and formula_49 are extrapolations of transmembrane potential and gating variables at previous timesteps with respect to formula_50, considering as many time instants as the order formula_51 of the BDF scheme. formula_52 is a coefficient which depends on the BDF order formula_51.
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:
formula_56
formula_59
Another possible segregated scheme would be the one in which formula_60 is calculated first, and then it is used in the equations for formula_54.
References.
 ;templatestyles src="Reflist/styles.css" / ;