# 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 = -I*_{ion }/C_{m} ,

where *C*_{m} = 2 μF/cm^{2} is cell capacitance per unit surface area.
**Membrane currents**

*I*_{ion} = I_{Na} + I_{K1} + I_{to}
+ I_{Kr} + I_{Ks} + I_{CaL} + I_{NaCa} + I_{NaK}
+ I_{pCa} + I_{pK} + I_{bCa} + I_{bNa}

where
*I*_{Na} is fast *Na*^{+} current,
*I*_{CaL} is L-type *Ca*^{ 2+} current,
*I*_{to} is transient outward current,
*I*_{Kr} is rapid delayed rectifier current,
*I*_{Ks} is slow delayed rectifier current,
*I*_{K1} is inward rectifier *K*^{+} current,
*I*_{NaCa} is *Na*^{+}/ Ca^{ 2+} exchanger current,
*I*_{NaK} is *Na*^{+}/ K^{+} pump current,
*I*_{pCa} and *I*_{pK} are plateau *Ca*^{ 2+}
and *K*^{+} currents, and *I*_{bCa} and
*I*_{bK} are background *Ca*^{ 2+} and *K*^{+} currents.

**Reversal Potentials**

*E*_{x} = (RT/zF) log(X_{o} / X_{i} )
for *X = Na*^{+}, K^{+}, Ca^{ 2+}

E_{Ks} = (RT/F) log[(K_{o} + p_{KNa} Na_{o} )
/ (K_{i} + p_{KNa} Na_{i} )]

**Fast ***Na*^{+} current *I*_{Na}.
The three gates formulation first introduced by Beeler and Reuter is used

*I*_{Na} = G_{Na} m^{3}h j(V - E_{Na} )

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
(*τ*_{m }, τ_{j }, τ_{h }),
which are functions of membrane potential *V*.

*m*_{∞ }(V),
h_{∞ }(V) ≡
j_{∞}(V) and
*7τ*_{m }(V),
τ_{h }(V)/50,
τ_{j }(V)/400 (ms)

are ploted for *-100 ≤ V ≤ 50 (mV)* (here
*0 ≤ y ≤ 1* ). Note, that *τ*_{m }(-85) = 0.0011 ms
is very small, therefore we have to use Rush and Larsen
integration scheme for temporal dynamics simulations.
*τ*_{h }(20) = 0.18 and *τ*_{j }(20) = 0.54 ms.

**L-type ***Ca*^{ 2+} current *I*_{CaL} [2]

*I*_{CaL} = G_{CaL} d f f_{2} f_{CaSS}
4 ((V - 15)F^{2}/RT) (0.25 Ca_{SS} e^{2(V - 15)F/RT} - Ca_{o} )
/ (e^{2(V - 15)F/RT} - 1),

f_{CaSS∞} = 0.4 + 0.6/(1 + (CaSS/0.05)^{2}),
τ_{CaSS} = 2 + 80/(1 + (CaSS/0.05)^{2}),

where *d* is a voltage-dependent activation gate, *f* is
a slow voltage inactivation gate, *f*_{2} is a fast voltage inactivation gate,
*f*_{CaSS } is an intracellular calcium-dependent inactivation gate.

*d*_{∞ },
f_{∞},
f_{2 ∞} and
*τ*_{d }/2,
τ_{f }/1500,
τ_{f2 }/600.

**Transient outward current ***I*_{to}

*I*_{to} = G_{to} r s (V - E_{K} )

where *r* is a voltage-dependent activation gate and *s*
is a voltagedependent inactivation gate.

*r*_{∞ },
s_{∞} (mid) and
*τ*_{r }/12,
τ_{s }/100.

**Slow delayed rectifier current ***I*_{Ks} [2]

*I*_{Ks} = G_{Ks} x_{s}^{2}
(V - E_{Ks} )

where *x*_{s} is an activation gate and
*E*_{Ks} is a reversal potential determined by a large permeability
to potassium and a small permeability to sodium ions.

**Inward rectifier ***K*^{+} current *I*_{K1}

*I*_{K1} = G_{K1} (K_{o} /
5.4)^{1/2}x_{K1 ∞} (V - E_{K} )

where *x*_{K1 ∞} is a time-independent inward
rectification factor that is a function of voltage *V - E*_{K} .

**Rapid delayed rectifier current ***I*_{Kr}

*I*_{Kr} = G_{Kr} (K_{o }/5.4)^{1/2}
x_{r1} x_{r2} (V - E_{K} )

where *x*_{r1} is an activation gate and *x*_{r2}
is an inactivation gate.

*x*_{r1 ∞ },
x_{r2 ∞},
x_{s ∞},
x_{K1 ∞} and
*τ*_{xr1 }/1250,
τ_{xr2 }/3.5,
τ_{xs }/1200.

*Na*^{+}/ Ca^{ 2+} exchanger current *I*_{NaCa}

*I*_{NaCa} = k_{NaCa}
(e ^{γ VF/RT}Na_{i}^{3} Ca_{o} -
e ^{(1 - γ) VF/RT}Na_{o}^{3} Ca_{i} α) /

[(K_{mNai}^{3} + Na_{o}^{3} )
(K_{mCa} + Ca_{o} )
(1 + k_{sat} e ^{(1 - γ) VF/RT})].

*Na*^{+}/ K^{+} pump current *I*_{NaK}

*I*_{NaK} = R_{NaK} K_{o}Na_{i}
/ [(K_{o} + K_{mK} )(Na_{i} + K_{mNa}
)(1 + 0.1245 e^{-11VF/RT} + 0.0353 e^{-VF/RT})].

**Intracellular ion dynamics**

The changes in the intracellular sodium *Na*_{i} and potassium
*K*_{i} concentrations are governed by the following equations

*dNa*_{i }/dt = -(I_{Na} + I_{bNa}
+ 3I_{NaK} + 3I_{NaCa} ) / V_{c }F

dK_{i} /dt = -(I_{K1} + I_{to} + I_{Kr}
+ I_{Ks} - 2I_{NaK} + I_{pK} + I_{stim} - I_{ax} )
/V_{c }F.

The model contains a description of calcium dynamics in the subspace *Ca*_{SS },
cytoplasm *Ca*_{i} and sarcoplasmic reticulum *Ca*_{SR}

*J*_{leak} = V_{leak} (Ca_{SR} - Ca_{i} ),

J_{up} = V_{max up}
/(1 + K_{up}^{2}/Ca_{i}^{2})

J_{rel} = V_{rel} O (Ca_{SR} - Ca_{SS} )

J_{xfer} = V_{xfer} (Ca_{SS} - Ca_{i} )

O = k_{1} Ca_{SS}^{2} R /
(k_{3} + k_{1} Ca_{SS}^{2}),

dR/dt = -k_{2} Ca_{SS} R + k_{4} (1 - R),
k_{1} = k_{1'}/k_{casr} ,
k_{2} = k_{2'} k_{casr} ,

Ca_{i buf c} = (Ca_{i} Bufc) / (Ca_{i} + Kbufc)

dCa_{i total} / dt = - (I_{bCa} + I_{pCa} -
2 I_{NaCa} )/2V_{c}F + (J_{leak} - J_{up} )
V_{sr }/V_{c} + J_{xfer}

Ca_{sr buf sr} = (Ca_{sr} Bu fsr) / (Ca_{sr} + Kbufsr)

dCa_{SRtotal }/dt = (J_{up} - J_{leak} - J_{rel} )

Ca_{SSbufSS} = (Ca_{ss} Bufss) / (Ca_{ss} + Kbufss)

dCa_{SStotal }/dt = - I_{CaL} /2V_{ss }F +
J_{rel} V_{sr} / V_{ss} -
J_{xfer} V_{c} / V_{ss}

where *J*_{leak} is a leakage current from the sarcoplasmic reticulum to the
cytoplasm, *J*_{up} is a pump current taking up calcium in the SR,
*J*_{rel} is the calcium-induced calcium release (CICR) current,
*Ca*_{itotal} is the total calcium in the cytoplasm,
it consists of *Ca*_{ibufc}, the buffered calcium in the cytoplasm,
and *Ca*_{i}, the free calcium in the cytoplasm. Similarly, *Ca*_{SR}
is the total calcium in the SR, it consists of *Ca*_{srbufsr}, the buffered
calcium in the SR, and *Ca*_{SR}, 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