Tusscher-Noble-Noble-Panfilov (TNNP) model

The cell membrane is modeled as a capacitor connected in parallel with variable resistances and batteries representing the different ionic currents and pumps
    dV/dt = -Iion /Cm ,
where Cm = 2 μF/cm2 is cell capacitance per unit surface area.

Membrane currents
    Iion = INa + IK1 + Ito + IKr + IKs + ICaL + INaCa + INaK + IpCa + IpK + IbCa + IbNa
where INa is fast Na+ current, ICaL is L-type Ca 2+ current, Ito is transient outward current, IKr is rapid delayed rectifier current, IKs is slow delayed rectifier current, IK1 is inward rectifier K+ current, INaCa is Na+/ Ca 2+ exchanger current, INaK is Na+/ K+ pump current, IpCa and IpK are plateau Ca 2+ and K+ currents, and IbCa and IbK are background Ca 2+ and K+ currents.

Reversal Potentials
    Ex = (RT/zF) log(Xo / Xi )     for X = Na+, K+, Ca 2+
    EKs = (RT/F) log[(Ko + pKNa Nao ) / (Ki + pKNa Nai )]

Fast Na+ current INa. The three gates formulation first introduced by Beeler and Reuter is used
    INa = GNa m3h j(V - ENa )
where m is an activation gate, h is a fast inactivation gate, and j is a slow inactivation gate. Each of these gates is governed by Hodgkin-Huxley-type equations for gating variables and characterized by a steady-state value (m, j, h) and a time constant for reaching this steady-state value (τm , τj , τh ), which are functions of membrane potential V.
m(V), h(V)j(V) and   7τm (V), τh (V)/50, τj (V)/400 (ms)

are ploted for -100 ≤ V ≤ 50 (mV) (here 0 ≤ y ≤ 1 ). Note, that τm (-85) = 0.0011 ms is very small, therefore we have to use Rush and Larsen integration scheme for temporal dynamics simulations. τh (20) = 0.18 and τj (20) = 0.54 ms.

L-type Ca 2+ current ICaL [2]
    ICaL = GCaL d f f2 fCaSS 4 ((V - 15)F2/RT) (0.25 CaSS e2(V - 15)F/RT - Cao ) / (e2(V - 15)F/RT - 1),
    fCaSS∞ = 0.4 + 0.6/(1 + (CaSS/0.05)2),   τCaSS = 2 + 80/(1 + (CaSS/0.05)2),

where d is a voltage-dependent activation gate, f is a slow voltage inactivation gate, f2 is a fast voltage inactivation gate, fCaSS is an intracellular calcium-dependent inactivation gate.
d, f, f2 ∞ and   τd /2, τf /1500, τf2 /600.

Transient outward current Ito
    Ito = Gto r s (V - EK )
where r is a voltage-dependent activation gate and s is a voltagedependent inactivation gate.
r, s (mid) and   τr /12, τs /100.

Slow delayed rectifier current IKs [2]
    IKs = GKs xs2 (V - EKs )
where xs is an activation gate and EKs is a reversal potential determined by a large permeability to potassium and a small permeability to sodium ions.

Inward rectifier K+ current IK1
    IK1 = GK1 (Ko / 5.4)1/2xK1 ∞ (V - EK )
where xK1 ∞ is a time-independent inward rectification factor that is a function of voltage V - EK .

Rapid delayed rectifier current IKr
    IKr = GKr (Ko /5.4)1/2 xr1 xr2 (V - EK )
where xr1 is an activation gate and xr2 is an inactivation gate.
xr1 ∞ , xr2 ∞, xs ∞, xK1 ∞ and   τxr1 /1250, τxr2 /3.5, τxs /1200.

Na+/ Ca 2+ exchanger current INaCa
    INaCa = kNaCa (e γ VF/RTNai3 Cao - e (1 - γ) VF/RTNao3 Cai α) /
      [(KmNai3 + Nao3 ) (KmCa + Cao ) (1 + ksat e (1 - γ) VF/RT)]
.

Na+/ K+ pump current INaK
    INaK = RNaK KoNai / [(Ko + KmK )(Nai + KmNa )(1 + 0.1245 e-11VF/RT + 0.0353 e-VF/RT)].

Intracellular ion dynamics

The changes in the intracellular sodium Nai and potassium Ki concentrations are governed by the following equations
    dNai /dt = -(INa + IbNa + 3INaK + 3INaCa ) / Vc F
    dKi /dt = -(IK1 + Ito + IKr + IKs - 2INaK + IpK + Istim - Iax ) /Vc F
.

The model contains a description of calcium dynamics in the subspace CaSS , cytoplasm Cai and sarcoplasmic reticulum CaSR
    Jleak = Vleak (CaSR - Cai ),
    Jup = Vmax up /(1 + Kup2/Cai2)
    Jrel = Vrel O (CaSR - CaSS )
    Jxfer = Vxfer (CaSS - Cai )
    O = k1 CaSS2 R / (k3 + k1 CaSS2),
    dR/dt = -k2 CaSS R + k4 (1 - R),     k1 = k1'/kcasr ,     k2 = k2' kcasr ,
    Cai buf c = (Cai Bufc) / (Cai + Kbufc)
    dCai total / dt = - (IbCa + IpCa - 2 INaCa )/2VcF + (Jleak - Jup ) Vsr /Vc + Jxfer
    Casr buf sr = (Casr Bu fsr) / (Casr + Kbufsr)
    dCaSRtotal /dt = (Jup - Jleak - Jrel )
    CaSSbufSS = (Cass Bufss) / (Cass + Kbufss)
    dCaSStotal /dt = - ICaL /2Vss F + Jrel Vsr / Vss - Jxfer Vc / Vss

where Jleak is a leakage current from the sarcoplasmic reticulum to the cytoplasm, Jup is a pump current taking up calcium in the SR, Jrel is the calcium-induced calcium release (CICR) current, Caitotal is the total calcium in the cytoplasm, it consists of Caibufc, the buffered calcium in the cytoplasm, and Cai, the free calcium in the cytoplasm. Similarly, CaSR is the total calcium in the SR, it consists of Casrbufsr, the buffered calcium in the SR, and CaSR, the free calcium in the SR.

The TNNP model scripts are based on Kirsten H.W.J. ten Tusscher's kirstennew2d.f codes (AJP 2006).

[1] Ten Tusscher K.H.W.J., D. Noble, P.J. Noble, and A.V. Panfilov.   A model for human ventricular tissue.   Am J Physiol Heart Circ Physiol 286 H1573 (2004).
[2] Ten Tusscher K.H., Panfilov A.V.   Alternans and spiral breakup in a human ventricular tissue model.   Am.J.Physiol., 90 326-345 (2006).


Heart rhythms     updated 9 May 2012