Bueno-Orovio–Cherry–Fenton model
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
The system of four differential equations reads as follows:[1]
- [math]\displaystyle{ \begin{cases} \dfrac{\partial u}{\partial t} = \nabla \cdot (D \nabla u) - ( J_{fi} + J_{so} + J_{si}) & \text{in } \Omega \times (0, T) \\[5pt] \dfrac{\partial v}{\partial t} = \dfrac{(1 - H(u - \theta_v)) (v_\infty - v)}{\tau_v^-} - \dfrac{H(u - \theta_v) v}{\tau_v^+} & \text{in } \Omega \times (0, T) \\[5pt] \dfrac{\partial w}{\partial t} = \dfrac{(1 - H(u - \theta_w)) (w_\infty - w)}{\tau_w^-} - \dfrac{H(u - \theta_w) w}{\tau_w^+} & \text{in } \Omega \times (0, T) \\[5pt] \dfrac{\partial s}{\partial t} = \dfrac{1}{\tau_s} \left( \dfrac{1 + \tanh(k_s (u - u_s))}{2}-s \right) & \text{in } \Omega \times (0, T) \\[5pt] \big( D \nabla u \big) \cdot \boldsymbol{N} = 0 & \text{on } \partial \Omega \times (0, T) \\[5pt] u = u_0, \, v = v_0, \, w = w_0, \, s = s_0 & \text{in } \Omega \times \{0\} \end{cases} }[/math]
where [math]\displaystyle{ \Omega }[/math] is the spatial domain and [math]\displaystyle{ T }[/math] is the final time. The initial conditions are [math]\displaystyle{ u_0 = 0 }[/math], [math]\displaystyle{ v_0 = 1 }[/math], [math]\displaystyle{ w_0 = 1 }[/math], [math]\displaystyle{ s_0 = 0 }[/math].[1]
[math]\displaystyle{ H(x - x_0) }[/math] refers to the Heaviside function centered in [math]\displaystyle{ x_0 }[/math]. The adimensional transmembrane potential [math]\displaystyle{ u }[/math] can be rescaled in mV by means of the affine transformation [math]\displaystyle{ V_{mV} = 85.7 u - 84 }[/math].[1] [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math] and [math]\displaystyle{ s }[/math] refer to gating variables, where in particular [math]\displaystyle{ s }[/math] can be also used as an indication of intracellular calcium [math]\displaystyle{ {{Ca}_i}^{2+} }[/math] concentration (given in the adimensional range [0, 1] instead of molar concentration).[5]
- [math]\displaystyle{ J_{fi}, J_{so} }[/math] and [math]\displaystyle{ J_{si} }[/math] are the fast inward, slow outward and slow inward currents respectively, given by the following expressions:[1]
- [math]\displaystyle{ J_{fi} = -\frac{v H(u - \theta_v) (u - \theta_v) (u_u - u)}{\tau_{fi}}, }[/math]
- [math]\displaystyle{ J_{so} = \frac{(u - u_o) (1 - H(u - \theta_w))}{\tau_o} + \frac{H(u - \theta_w)}{\tau_{so}}, }[/math]
- [math]\displaystyle{ J_{si} = -\frac{H(u - \theta_w) w s}{\tau_{si}}, }[/math]
All the above-mentioned ionic density currents are partially adimensional and are expressed in [math]\displaystyle{ \dfrac{1}{\text{seconds}} }[/math].[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]
- [math]\displaystyle{ \tau_v^- = (1 - H(u - \theta_v^-)) \tau_{v_1}^- + H(u - \theta_v^-) \tau_{v_2}^-, }[/math]
- [math]\displaystyle{ \tau_w^- = \tau_{w_1}^- + (\tau_{w_2}^- - \tau_{w_1}^-)(1 + \tanh(k_w^-(u - u_w^-)))/2, }[/math]
- [math]\displaystyle{ \tau_{so} = \tau_{{so}_1} + (\tau_{{so}_2} - \tau_{{so}_1})(1 + \tanh(k_{so} (u - u_{so})))/2, }[/math]
- [math]\displaystyle{ \tau_s = (1 - H(u - \theta_w)) \tau_{s_1} + H(u - \theta_w) \tau_{s_2}, }[/math]
- [math]\displaystyle{ \tau_o = (1 - H(u - \theta_o)) \tau_{o_1} + H(u - \theta_o) \tau_{o_2}, }[/math]
- [math]\displaystyle{ v_{\infty} = 1 - H(u - \theta_v^-), }[/math]
- [math]\displaystyle{ w_\infty = (1 - H(u - \theta_o))(1 - u/\tau_{w_\infty}) + H(u - \theta_o) w_\infty^*. }[/math]
where the temporal constants, i.e. [math]\displaystyle{ \tau_v^-, \tau_w^-, \tau_{so}, \tau_s, \tau_o }[/math] are expressed in seconds, whereas [math]\displaystyle{ v_{\infty} }[/math] and [math]\displaystyle{ w_{\infty} }[/math] are adimensional.[1]
The diffusion coefficient [math]\displaystyle{ D }[/math] results in a value of [math]\displaystyle{ 1.171 \pm 0.221 \dfrac{\text{cm}^2}{\text{seconds}} }[/math], 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 [math]\displaystyle{ \Omega }[/math], a forcing term [math]\displaystyle{ J_\text{app}(\boldsymbol x, t) }[/math], 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]
Parameter | [math]\displaystyle{ u_o }[/math] | [math]\displaystyle{ u_u }[/math] | [math]\displaystyle{ \theta_v }[/math] | [math]\displaystyle{ \theta_w }[/math] | [math]\displaystyle{ \theta_v^- }[/math] | [math]\displaystyle{ \theta_o }[/math] | [math]\displaystyle{ \tau_{v_1}^- }[/math] | [math]\displaystyle{ \tau_{v_2}^- }[/math] | [math]\displaystyle{ \tau_{v}^+ }[/math] | [math]\displaystyle{ \tau_{w_1}^- }[/math] | [math]\displaystyle{ \tau_{w_2}^- }[/math] | [math]\displaystyle{ k_w^- }[/math] | [math]\displaystyle{ u_w^- }[/math] | [math]\displaystyle{ \tau_w^+ }[/math] | [math]\displaystyle{ \tau_{fi} }[/math] | [math]\displaystyle{ \tau_{o_1} }[/math] | [math]\displaystyle{ \tau_{o_2} }[/math] | [math]\displaystyle{ \tau_{{so}_1} }[/math] | [math]\displaystyle{ \tau_{{so}_2} }[/math] | [math]\displaystyle{ k_{so} }[/math] | [math]\displaystyle{ u_{so} }[/math] | [math]\displaystyle{ \tau_{s_1} }[/math] | [math]\displaystyle{ \tau_{s_2} }[/math] | [math]\displaystyle{ k_s }[/math] | [math]\displaystyle{ u_s }[/math] | [math]\displaystyle{ \tau_{si} }[/math] | [math]\displaystyle{ \tau_{w_{\infty}} }[/math] | [math]\displaystyle{ w_{\infty}^* }[/math] |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
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
Assume that [math]\displaystyle{ \boldsymbol{z} }[/math] refers to the vector containing all the gating variables, i.e. [math]\displaystyle{ \boldsymbol{z} = [v, w, s]^T }[/math], and [math]\displaystyle{ \boldsymbol{F}:\mathbb{R}^4 \rightarrow \mathbb{R}^3 }[/math] 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]
- [math]\displaystyle{ \begin{cases} \dfrac{\partial u}{\partial t} - \nabla \cdot (D \nabla u) + ( J_{fi} + J_{so} + J_{si} ) = 0 & \text{in } \Omega \times (0, T) \\[5pt] \dfrac{\partial \boldsymbol{z}}{\partial t} = \boldsymbol{F}(u, \boldsymbol{z}) & \text{in } \Omega \times (0, T) \\ \end{cases} }[/math]
Let [math]\displaystyle{ p \in U = H^1(\Omega) }[/math] and [math]\displaystyle{ \boldsymbol{q} \in \boldsymbol{W}=[L^2(\Omega)]^3 }[/math] be two generic test functions.[6]
To obtain the weak formulation:[6]
- multiply by [math]\displaystyle{ p \in U }[/math] the first equation of the model and by [math]\displaystyle{ \boldsymbol{q} \in \boldsymbol{W} }[/math] the equations for the evolution of the gating variables. Integrating both members of all the equations in the domain [math]\displaystyle{ \Omega }[/math]:[6]
- [math]\displaystyle{ \begin{cases} \displaystyle \int_\Omega \dfrac{du(t)}{dt} p \,d\Omega - \int_\Omega \nabla \cdot (D \nabla u(t)) p \,d\Omega + \int_\Omega (J_{fi} + J_{so} + J_{si}) p \,d\Omega = 0 & \forall p \in U \\[5pt] \displaystyle \int_\Omega \dfrac{d\boldsymbol{z}(t)}{dt} \boldsymbol{q} \,d\Omega = \int_\Omega \boldsymbol{F}(u(t), \boldsymbol{z}(t)) \boldsymbol{q} \,d\Omega & \forall \boldsymbol{q} \in \boldsymbol{W} \end{cases} }[/math]
- perform an integration by parts of the diffusive term of the first monodomain-like equation:[6]
- [math]\displaystyle{ - \int_\Omega \nabla \cdot (D \nabla u(t)) p \,d\Omega = \int_\Omega D \nabla u(t) \nabla p \, d\Omega - \cancelto{\text{Neumann B.C.}}{\int_{\partial \Omega} (D \nabla u(t)) \cdot \boldsymbol{N} p \,d\Omega} }[/math]
Finally the weak formulation reads:
- Find [math]\displaystyle{ u \in L^2(0, T; H^1(\Omega)) }[/math] and [math]\displaystyle{ \boldsymbol{z} \in L^2(0, T; [L^2(\Omega)]^3) }[/math], [math]\displaystyle{ \forall t \in (0, T) }[/math], such that[6]
- [math]\displaystyle{ \begin{cases} \displaystyle \int_\Omega \frac{du(t)}{dt} p \,d\Omega + \int_{\Omega} D \nabla u(t) \cdot \nabla p \,d\Omega + \int_\Omega (J_{fi} + J_{so} + J_{si}) p \,d\Omega = 0 & \forall p \in U \\[5pt] \displaystyle \int_\Omega \frac{d\boldsymbol{z}(t)}{dt} \boldsymbol{q} \,d\Omega = \int_\Omega \boldsymbol{F}(u(t), \boldsymbol{z}(t)) \boldsymbol{q} \,d\Omega & \forall \boldsymbol{q} \in \boldsymbol{W} \\[5pt] u(0) = u_0 \\[5pt] \boldsymbol{z}(0) = \boldsymbol{z}_0 \end{cases} }[/math]
Numerical discretization
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 [math]\displaystyle{ \sigma }[/math] or a Runge–Kutta method (RK).[7][5]
Space discretization with FEM
Let [math]\displaystyle{ \mathcal{T}_h }[/math] be a tessellation of the computational domain [math]\displaystyle{ \Omega }[/math] by means of a certain type of elements (such as tetrahedrons or hexahedra), with [math]\displaystyle{ h }[/math] representing a chosen measure of the size of a single element [math]\displaystyle{ K \in \mathcal{T}_h }[/math]. Consider the set [math]\displaystyle{ \mathbb{P}^r(K) }[/math] of polynomials with degree smaller or equal than [math]\displaystyle{ r }[/math] over an element [math]\displaystyle{ K }[/math]. Define [math]\displaystyle{ \mathcal{X}_h^r = \{f \in C^0(\bar \Omega): f|_K \in \mathbb{P}^r(K) \,\, \forall K \in \mathcal{T}_h \} }[/math] as the finite dimensional space, whose dimension is [math]\displaystyle{ N_h=\dim(\mathcal{X}_h^r) }[/math]. The set of basis functions of [math]\displaystyle{ \mathcal{X}_h^r }[/math] is referred to as [math]\displaystyle{ \{ \phi_i \}_{i=1}^{N_h} }[/math].[5]
The semidiscretized formulation of the first equation of the model reads: find [math]\displaystyle{ u_h = u_h(t) = \sum_{j=1}^{N_h} \bar{u}_j(t) \phi_j }[/math] projection of the solution [math]\displaystyle{ u(t) }[/math] on [math]\displaystyle{ \mathcal{X}_{h}^r }[/math], [math]\displaystyle{ \forall t \in (0, T) }[/math], such that[5]
- [math]\displaystyle{ \int_\Omega {\dot{u}}_{h} \phi_i \, d \Omega + \int_\Omega (D \nabla u_h ) \cdot \nabla \phi_i \, d\Omega + \int_\Omega J_\text{ion}(u_h, \boldsymbol{z}_h) \phi_i \, d \Omega = 0 \quad \text{for } i = 1, \ldots, N_h }[/math]
with [math]\displaystyle{ u_h(0) = \sum_{j=1}^{N_h} \left( \int_\Omega u_0 \phi_j \,d\Omega \right) \phi_j }[/math], [math]\displaystyle{ \boldsymbol{z}_{h} = \boldsymbol{z}_h(t) = [v_h(t), w_h(t), s_h(t)]^T }[/math] semidiscretized version of the three gating variables, and [math]\displaystyle{ J_\text{ion}(u_h, \boldsymbol{z}_{h})=J_{fi}(u_h, \boldsymbol{z}_h) + J_{so}(u_h, \boldsymbol{z}_h) + J_{si}(u_h, \boldsymbol{z}_h) }[/math] 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 [math]\displaystyle{ \boldsymbol{U}_h(t) = \{ \bar{u}_j(t) \}_{j=1}^{N_h} }[/math] and [math]\displaystyle{ \boldsymbol{Z}_h(t) = \{ \bar{\boldsymbol{z}}_j(t) \}_{j=1}^{N_h} }[/math]:[5]
- [math]\displaystyle{ \begin{cases} \mathbb{M} \dot{\boldsymbol{U}}_h(t) + \mathbb{K} \boldsymbol{U}_h(t) + \boldsymbol{J}_\text{ion}(\boldsymbol{U}_h(t), \boldsymbol{Z}_h(t)) = 0 & \forall t \in (0, T) \\ \boldsymbol{U}_h(0) = \boldsymbol{U}_{0, h} \end{cases} }[/math]
where [math]\displaystyle{ \mathbb{M}_{ij} = \int_\Omega \phi_j \phi_i \,d \Omega }[/math], [math]\displaystyle{ \mathbb{K}_{ij} = \int_\Omega D \nabla \phi_j \cdot \nabla \phi_i \,d \Omega }[/math] and [math]\displaystyle{ \left( \boldsymbol{J}_\text{ion}(\boldsymbol{U}_{h}(t), \boldsymbol{z}_{h}(t)) \right)_i = \int_\Omega J_\text{ion}(u_h, \boldsymbol{z}_{h}) \phi_i \,d \Omega }[/math].[5]
The non-linear term [math]\displaystyle{ \boldsymbol{J}_\text{ion}(\boldsymbol{U}_{h}(t), \boldsymbol{Z}_{h}(t)) }[/math] 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 [math]\displaystyle{ \{ \boldsymbol{x}_s^K \}_{s=1}^{N_q} }[/math] and [math]\displaystyle{ \{ \omega_s^K \}_{s=1}^{N_q} }[/math] the quadrature nodes and weights of a generic element of the mesh [math]\displaystyle{ K \in \mathcal{T}_{h} }[/math], both [math]\displaystyle{ u_h }[/math] and [math]\displaystyle{ \boldsymbol{z}_h }[/math] are evaluated at the quadrature nodes:[5]
- [math]\displaystyle{ \int_\Omega J_\text{ion}(u_h, \boldsymbol{z}_h) \phi_i \, d \Omega \approx \sum_{K \in \mathcal{T}_h} \left( \sum_{s=1}^{N_q} J_\text{ion} \left( \sum_{j=1}^{N_h} \bar{u}_j(t) \phi_j(\boldsymbol{x}_s^K), \sum_{j=1}^{N_h} \bar{\boldsymbol{z}}_j(t) \phi_j(\boldsymbol{x}_s^K) \right) \phi_i(\boldsymbol{x}_s^K) \omega_s^K \right) }[/math]
The equations for the three gating variables, which are ODEs, are directly solved in all the degrees of freedom (DOF) of the tessellation [math]\displaystyle{ \mathcal{T}_h }[/math] separately, leading to the following semidiscrete form:[5]
- [math]\displaystyle{ \begin{cases} \dot{\boldsymbol{Z}}_h(t) = \boldsymbol{F}(\boldsymbol{U}_h(t), \boldsymbol{Z}_h(t)) & \forall t \in (0, T) \\ \boldsymbol{Z}_h(0) = \boldsymbol{Z}_{0, h} \end{cases} }[/math]
Time discretization with BDF (implicit scheme)
With reference to the time interval [math]\displaystyle{ (0, T] }[/math], let [math]\displaystyle{ \Delta t = \dfrac{T}{N} }[/math] be the chosen time step, with [math]\displaystyle{ N }[/math] number of subintervals. A uniform partition in time [math]\displaystyle{ [t_0 = 0, t_1 = \Delta t, \ldots, t_k, \ldots, t_{N-1}, t_N = T] }[/math] 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 [math]\displaystyle{ t = t^k }[/math], the full problem is entirely solved in one step in order to get [math]\displaystyle{ (\boldsymbol{U}_h^{k + 1}, \boldsymbol{Z}_h^{k + 1}) }[/math] by means of either Newton method or Fixed-point iterations:[11]
- [math]\displaystyle{ \begin{cases} \mathbb{M} \alpha \dfrac{ \boldsymbol{U}_{h}^{k + 1} - \boldsymbol{U}_{\text{ext}, h}^{k}}{\Delta t} + \mathbb{K} \boldsymbol{U}_h^{k + 1} + \boldsymbol{J}_\text{ion}(\boldsymbol{U}_{h}^{k + 1}, \boldsymbol{Z}_{h}^{k + 1}) = 0 \\ \alpha \dfrac{\boldsymbol{Z}_h^{k + 1} - \boldsymbol{Z}_{\text{ext}, h}^{k}}{\Delta t} = \boldsymbol{F}(\boldsymbol{U}_h^{k + 1}, \boldsymbol{Z}_h^{k + 1} ) \end{cases} }[/math]
where [math]\displaystyle{ \boldsymbol{U}_{\text{ext}, h}^{k} }[/math] and [math]\displaystyle{ \boldsymbol{Z}_{\text{ext}, h}^{k} }[/math] are extrapolations of transmembrane potential and gating variables at previous timesteps with respect to [math]\displaystyle{ t^{k+1} }[/math], considering as many time instants as the order [math]\displaystyle{ \sigma }[/math] of the BDF scheme. [math]\displaystyle{ \alpha }[/math] is a coefficient which depends on the BDF order [math]\displaystyle{ \sigma }[/math].[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, [math]\displaystyle{ \boldsymbol{Z}_h^{k + 1} }[/math] is calculated, using an extrapolation at previous timesteps [math]\displaystyle{ \boldsymbol{U}_{\text{ext}, h}^k }[/math] for the transmembrane potential at the right hand side:[11]
- [math]\displaystyle{ \alpha \dfrac{\boldsymbol{Z}_h^{k + 1} - \boldsymbol{Z}_{\text{ext}, h}^{k}}{\Delta t} = \boldsymbol{F}(\boldsymbol{U}_{\text{ext}, h}^k, \boldsymbol{Z}_h^{k + 1} ) }[/math]
- Secondly, [math]\displaystyle{ \boldsymbol{U}_h^{k + 1} }[/math] is computed, exploiting the value of [math]\displaystyle{ \boldsymbol{Z}_h^{k + 1} }[/math] that has just been calculated:[11]
- [math]\displaystyle{ \mathbb{M} \alpha \dfrac{ \boldsymbol{U}_h^{k + 1} - \boldsymbol{U}_{\text{ext}, h}^k}{\Delta t} + \mathbb{K} \boldsymbol{U}_h^{k + 1} + \boldsymbol{J}_\text{ion}(\boldsymbol{U}_h^{k + 1}, \boldsymbol{Z}_h^{k + 1}) = 0 }[/math]
Another possible segregated scheme would be the one in which [math]\displaystyle{ \boldsymbol{U}_{h}^{k + 1} }[/math] is calculated first, and then it is used in the equations for [math]\displaystyle{ \boldsymbol{Z}_h^{k + 1} }[/math].[11]
See also
References
- ↑ 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10 1.11 1.12 1.13 1.14 1.15 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.
- ↑ 2.0 2.1 Colli Franzone, P.; Pavarino, L.F.; Scacchi, S. (30 October 2014). Mathematical cardiac electrophysiology. ISBN 978-3-319-04801-7.
- ↑ 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.
- ↑ Keener, J.; Sneyd, J. (27 October 2008). Mathematical physiology (2nd ed.). Springer. ISBN 978-0-387-79387-0.
- ↑ 5.00 5.01 5.02 5.03 5.04 5.05 5.06 5.07 5.08 5.09 5.10 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.
- ↑ 6.0 6.1 6.2 6.3 6.4 6.5 6.6 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". Computer Methods in Applied Mechanics and Engineering 343: 52–73. doi:10.1016/j.cma.2018.08.032. Bibcode: 2019CMAME.343...52P. https://re.public.polimi.it/bitstream/11311/1066014/2/1-s2.0-S0045782518304304-main.pdf.
- ↑ 7.0 7.1 7.2 Quarteroni, A. (25 April 2014). Numerical models for differential problems (Second ed.). ISBN 978-88-470-5522-3.
- ↑ 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.
- ↑ 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.
- ↑ 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.
- ↑ 11.0 11.1 11.2 11.3 11.4 11.5 11.6 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.
Original source: https://en.wikipedia.org/wiki/Bueno-Orovio–Cherry–Fenton model.
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