Ten Tusscher et al. 2006¶

Key: tentusscher2006alternans

This detailed human ventricular cell model was developed to investigate the mechanisms of electrical instability, alternans, and spiral wave breakup in cardiac tissue. It builds on their 2004 model by incorporating a more comprehensive description of intracellular calcium dynamics, including subspace calcium compartments and a Markov model of the ryanodine receptor for calcium-induced calcium release (CICR). The model also includes both fast and slow voltage-dependent inactivation gates for the L-type calcium current, enabling more accurate simulation of calcium handling and its role in arrhythmogenesis. It reproduces a wide range of experimentally observed APD restitution slopes (see the parameter sets slope, default slope[1.1]) and explores the interaction between sodium current recovery dynamics and tissue-level instability. The model is suited for studying reentry, alternans, and ventricular fibrillation mechanisms in human cardiac tissue. The model also has different parameter sets for epicardial (parameter set epi, default), endocardial (endo), and midmyocardial cells (m).

References¶

Figure¶

Variables¶

V = -85.23
Ca_SR = 3.64
Ca_i = 0.000126
Ca_ss = 0.00036
K_i = 136.89
Na_i = 8.604
Rbar = 0.9073
Xr1 = 0.00621
Xr2 = 0.4712
Xs = 0.0095
d = 3.373e-05
f2 = 0.9755
f = 0.7888
fCass = 0.9953
h = 0.7444
j = 0.7045
m = 0.00172
r = 2.42e-08
s = 0.999998

Parameters¶

diffusivity_V = 1.0
Buf_c = 0.2
Buf_sr = 10.0
Buf_ss = 0.4
C_m = 185.0
Ca_o = 2.0
EC = 1.5
F = 96.485
K_buf_c = 0.001
K_buf_sr = 0.3
K_buf_ss = 0.00025
K_mCa = 1.38
K_mNa = 40.0
K_mNa_i = 87.5
K_mk = 1.0
K_o = 5.4
K_pCa = 0.0005
K_up = 0.00025
Na_o = 140.0
P_NaK = 2.724
R = 8.314
T = 310.0
V_c = 16404.0
V_leak = 0.00036
V_maxup = 0.006375
V_rel = 0.102
V_sr = 1094.0
V_ss = 54.68
V_xfer = 0.0038
alpha = 2.5
factor_f_tau = 1.0
factor_g_Kr = 1.0
factor_g_Ks = 1.0
factor_g_pCa = 1.0
factor_g_pK = 1.0
g_CaL = 0.0398
g_K1 = 5.405
g_Kr = 0.153
g_Ks = 0.392
g_Na = 14.838
g_bCa = 0.000592
g_bNa = 0.00029
g_pCa = 0.1238
g_pK = 0.0146
g_to = 0.294
gamma = 0.35
k1_prime = 0.15
k2_prime = 0.045
k3 = 0.06
k4 = 0.005
k_NaCa = 1000.0
k_sat = 0.1
max_sr = 2.5
min_sr = 1.0
p_KNa = 0.03
s_offset = 20.0
s_variant = 0.0

Source code¶
OpenCL kernel// i_CaL: gating variable d
const Real d_alpha = 1.4 / (1. + exp((-35. - V) / 13.)) + 0.25;
const Real d_beta = 1.4 / (1. + exp((V + 5.) / 5.));
const Real d_inf = 1. / (1. + exp((-8. - V) / 7.5));
const Real d_gamma = 1. / (1. + exp((50. - V) / 20.));
const Real d_tau = 1. * d_alpha * d_beta + d_gamma;
*_new_d = d_inf + (d - d_inf) * exp(-(dt / d_tau));

// i_CaL: gating variable f2
const Real f2_inf = 0.67 / (1. + exp((V + 35.) / 7.)) + 0.33;
const Real f2_tau = 562. * exp(-pow(V + 27., 2.) / 240.) + 31. / (1. + exp((25. - V) / 10.)) + 80. / (1. + exp((V + 30.) / 10.));
*_new_f2 = f2_inf + (f2 - f2_inf) * exp(-(dt / f2_tau));

// i_CaL: gating variable fCass
const Real fCass_inf = 0.6 / (1. + pow(Ca_ss / 0.05, 2.)) + 0.4;
const Real fCass_tau = 80. / (1. + pow(Ca_ss / 0.05, 2.)) + 2.;
*_new_fCass = fCass_inf + (fCass - fCass_inf) * exp(-(dt / fCass_tau));

// i_CaL: gating variable f
const Real f_inf = 1. / (1. + exp((V + 20.) / 7.));
const Real f_tau = 1102.5 * exp(-pow(V + 27., 2.) / 225.) + 200. / (1. + exp((13. - V) / 10.)) + 180. / (1. + exp((V + 30.) / 10.)) + 20.;
*_new_f = f_inf + (f - f_inf) * exp(-(dt / (factor_f_tau * f_tau)));

// i_pCa
const Real i_pCa = factor_g_pCa * g_pCa * Ca_i / (Ca_i + K_pCa);

// i_Na: gating variable h
const Real h_alpha = ((V < -40.) ? 0.057 * exp(-(V + 80.) / 6.8) : 0.);
const Real h_beta = ((V < -40.) ? 2.7 * exp(0.079 * V) + 310000. * exp(0.3485 * V) : 0.77 / (0.13 * (1. + exp((V + 10.66) / -11.1))));
const Real h_inf = 1. / (pow(1. + exp((V + 71.55) / 7.43), 2.));
const Real h_tau = 1. / (h_alpha + h_beta);
*_new_h = h_inf + (h - h_inf) * exp(-(dt / h_tau));

// i_Na: gating variable j
const Real j_alpha = ((V < -40.) ? (-25428. * exp(0.2444 * V) - 6.948e-06 * exp(-0.04391 * V)) * (V + 37.78) / 1. / (1. + exp(0.311 * (V + 79.23))) : 0.);
const Real j_beta = ((V < -40.) ? 0.02424 * exp(-0.01052 * V) / (1. + exp(-0.1378 * (V + 40.14))) : 0.6 * exp(0.057 * V) / (1. + exp(-0.1 * (V + 32.))));
const Real j_inf = 1. / (pow(1. + exp((V + 71.55) / 7.43), 2.));
const Real j_tau = 1. / (j_alpha + j_beta);
*_new_j = j_inf + (j - j_inf) * exp(-(dt / j_tau));

// i_Na: gating variable m
const Real m_alpha = 1. / (1. + exp((-60. - V) / 5.));
const Real m_beta = 0.1 / (1. + exp((V + 35.) / 5.)) + 0.1 / (1. + exp((V - 50.) / 200.));
const Real m_inf = 1. / (pow(1. + exp((-56.86 - V) / 9.03), 2.));
const Real m_tau = 1. * m_alpha * m_beta;
*_new_m = m_inf + (m - m_inf) * exp(-(dt / m_tau));

// i_Kr: gating variable Xr1
const Real Xr1_alpha = 450. / (1. + exp((-45. - V) / 10.));
const Real Xr1_beta = 6. / (1. + exp((V + 30.) / 11.5));
const Real Xr1_inf = 1. / (1. + exp((-26. - V) / 7.));
const Real Xr1_tau = 1. * Xr1_alpha * Xr1_beta;
*_new_Xr1 = Xr1_inf + (Xr1 - Xr1_inf) * exp(-(dt / Xr1_tau));

// i_Kr: gating variable Xr2
const Real Xr2_alpha = 3. / (1. + exp((-60. - V) / 20.));
const Real Xr2_beta = 1.12 / (1. + exp((V - 60.) / 20.));
const Real Xr2_inf = 1. / (1. + exp((V + 88.) / 24.));
const Real Xr2_tau = 1. * Xr2_alpha * Xr2_beta;
*_new_Xr2 = Xr2_inf + (Xr2 - Xr2_inf) * exp(-(dt / Xr2_tau));

// i_Ks: gating variable Xs
const Real Xs_alpha = 1400. / (sqrt(1. + exp((5. - V) / 6.)));
const Real Xs_beta = 1. / (1. + exp((V - 35.) / 15.));
const Real Xs_inf = 1. / (1. + exp((-5. - V) / 14.));
const Real Xs_tau = 1. * Xs_alpha * Xs_beta + 80.;
*_new_Xs = Xs_inf + (Xs - Xs_inf) * exp(-(dt / Xs_tau));

// i_to: gating variable r
const Real r_inf = 1. / (1. + exp((20. - V) / 6.));
const Real r_tau = 9.5 * exp(-pow(V + 40., 2.) / 1800.) + 0.8;
*_new_r = r_inf + (r - r_inf) * exp(-(dt / r_tau));

// i_to: gating variable s
const Real s_inf = 1. / (1. + exp((V + s_offset) / 5.));
Real s_tau = 0;
if(s_variant > 0.5) {
    s_tau = 1000. * exp(-pow(V + 67., 2.) / 1000.) + 8.;
} else {
    s_tau = 85. * exp(-pow(V + 45., 2.) / 320.) + 5. / (1. + exp((V - 20.) / 5.)) + 3.;
}
*_new_s = s_inf + (s - s_inf) * exp(-(dt / s_tau));

// dynCa
const Real f_JCa_i_free = 1. / (1. + Buf_c * K_buf_c / (pow(Ca_i + K_buf_c, 2.)));
const Real f_JCa_sr_free = 1. / (1. + Buf_sr * K_buf_sr / (pow(Ca_SR + K_buf_sr, 2.)));
const Real f_JCa_ss_free = 1. / (1. + Buf_ss * K_buf_ss / (pow(Ca_ss + K_buf_ss, 2.)));
const Real i_leak = V_leak * (Ca_SR - Ca_i);
const Real i_up = V_maxup / (1. + pow(K_up, 2.) / (pow(Ca_i, 2.)));
const Real i_xfer = V_xfer * (Ca_ss - Ca_i);
const Real kcasr = max_sr - (max_sr - min_sr) / (1. + pow(EC / (Ca_SR), 2.));
const Real k1 = k1_prime / (kcasr);
const Real k2 = k2_prime * kcasr;
const Real O = k1 * pow(Ca_ss, 2.) * Rbar / (k3 + k1 * pow(Ca_ss, 2.));
*_new_Rbar = Rbar + dt*(-k2 * Ca_ss * Rbar + k4 * (1. - Rbar));
const Real i_rel = V_rel * O * (Ca_SR - Ca_ss);
const Real ddt_Ca_sr_total = i_up - (i_rel + i_leak);
*_new_Ca_SR = Ca_SR + dt*(ddt_Ca_sr_total * f_JCa_sr_free);

// *remaining*
const Real E_Ca = 0.5 * R * T / F * log(Ca_o / (Ca_i));
const Real E_K = R * T / F * log(K_o / (K_i));
const Real E_Ks = R * T / F * log((K_o + p_KNa * Na_o) / (K_i + p_KNa * Na_i));
const Real E_Na = R * T / F * log(Na_o / (Na_i));
const Real i_CaL = g_CaL * d * f * f2 * fCass * 4. * (V - 15.) * pow(F, 2.) / (R * T) * (0.25 * Ca_ss * exp(2. * (V - 15.) * F / (R * T)) - Ca_o) / ((fabs(exp(2. * (V - 15.) * F / (R * T)) - 1.) < VERY_SMALL_NUMBER) ? ((exp(2. * (V - 15.) * F / (R * T)) - 1. < 0.) ? -VERY_SMALL_NUMBER : VERY_SMALL_NUMBER) : exp(2. * (V - 15.) * F / (R * T)) - 1.);
const Real i_NaK = P_NaK * K_o / (K_o + K_mk) * Na_i / (Na_i + K_mNa) / (1. + 0.1245 * exp(-0.1 * V * F / (R * T)) + 0.0353 * exp(-V * F / (R * T)));
const Real i_NaCa = k_NaCa * (exp(gamma * V * F / (R * T)) * pow(Na_i, 3.) * Ca_o - exp((gamma - 1.) * V * F / (R * T)) * pow(Na_o, 3.) * Ca_i * alpha) / ((pow(K_mNa_i, 3.) + pow(Na_o, 3.)) * (K_mCa + Ca_o) * (1. + k_sat * exp((gamma - 1.) * V * F / (R * T))));
const Real i_K1_alpha_K1 = 0.1 / (1. + exp(0.06 * (V - E_K - 200.)));
const Real i_K1_beta_K1 = (3. * exp(0.0002 * (V - E_K + 100.)) + exp(0.1 * (V - E_K - 10.))) / (1. + exp(-0.5 * (V - E_K)));
const Real i_Kr = factor_g_Kr * g_Kr * sqrt(K_o / 5.4) * Xr1 * Xr2 * (V - E_K);
const Real i_Ks = factor_g_Ks * g_Ks * pow(Xs, 2.) * (V - E_Ks);
const Real i_Na = g_Na * pow(m, 3.) * h * j * (V - E_Na);
const Real i_bNa = g_bNa * (V - E_Na);
const Real i_bCa = g_bCa * (V - E_Ca);
const Real i_to = g_to * r * s * (V - E_K);
const Real i_pK = factor_g_pK * g_pK * (V - E_K) / (1. + exp((25. - V) / 5.98));
const Real ddt_Ca_ss_total = -i_CaL * C_m / (2. * V_ss * F) + i_rel * V_sr / V_ss - i_xfer * V_c / V_ss;
const Real i_K1_xK1_inf = i_K1_alpha_K1 / (i_K1_alpha_K1 + i_K1_beta_K1);
const Real ddt_Ca_i_total = -(i_bCa + i_pCa - 2. * i_NaCa) * C_m / (2. * V_c * F) + (i_leak - i_up) * V_sr / V_c + i_xfer;
*_new_Ca_ss = Ca_ss + dt*(ddt_Ca_ss_total * f_JCa_ss_free);
*_new_Na_i = Na_i + dt*(-(i_Na + i_bNa + 3. * i_NaK + 3. * i_NaCa) / (V_c * F) * C_m);
const Real i_K1 = g_K1 * i_K1_xK1_inf * sqrt(K_o / 5.4) * (V - E_K);
*_new_Ca_i = Ca_i + dt*(ddt_Ca_i_total * f_JCa_i_free);
*_new_V = V + dt*(-(i_K1 + i_to + i_Kr + i_Ks + i_CaL + i_NaK + i_Na + i_bNa + i_NaCa + i_bCa + i_pK + i_pCa) + _diffuse_V);
*_new_K_i = K_i + dt*(-(i_K1 + i_to + i_Kr + i_Ks + i_pK - 2. * i_NaK) / (V_c * F) * C_m);

Additional metadata¶

keywords:
- excitable media
- electrophysiology
- heart
- human
- ventricle
initial values:
  epi:
    V: -85.23
    Ca_SR: 3.64
    Ca_i: 0.000126
    Ca_ss: 0.00036
    K_i: 136.89
    Na_i: 8.604
    Rbar: 0.9073
    Xr1: 0.00621
    Xr2: 0.4712
    Xs: 0.0095
    d: 3.373e-05
    f2: 0.9755
    f: 0.7888
    fCass: 0.9953
    h: 0.7444
    j: 0.7045
    m: 0.00172
    r: 2.42e-08
    s: 0.999998
  endo:
    V: -86.709
    Ca_SR: 3.715
    Ca_i: 0.00013
    Ca_ss: 0.00036
    K_i: 138.4
    Na_i: 10.355
    Rbar: 0.9068
    Xr1: 0.00448
    Xr2: 0.476
    Xs: 0.0087
    d: 3.164e-05
    f2: 0.9778
    f: 0.8009
    fCass: 0.9953
    h: 0.7573
    j: 0.7225
    m: 0.00155
    r: 2.235e-08
    s: 0.3212
  m:
    V: -85.423
    Ca_SR: 4.272
    Ca_i: 0.000153
    Ca_ss: 0.00042
    K_i: 138.52
    Na_i: 10.132
    Rbar: 0.8978
    Xr1: 0.0165
    Xr2: 0.473
    Xs: 0.0174
    d: 3.288e-05
    f2: 0.9526
    f: 0.7026
    fCass: 0.9942
    h: 0.749
    j: 0.6788
    m: 0.00165
    r: 2.347e-08
    s: 0.999998
parameter sets:
  slope:
    0.7:
      factor_g_Kr: 0.8758169934640524
      factor_g_Ks: 0.6887755102040817
      factor_g_pCa: 0.5
      factor_g_pK: 5.0
      factor_f_tau: 0.6
    1.1:
      factor_g_Kr: 1.0
      factor_g_Ks: 1.0
      factor_g_pCa: 1.0
      factor_g_pK: 1.0
      factor_f_tau: 1.0
    1.4:
      factor_g_Kr: 1.1241830065359477
      factor_g_Ks: 1.125
      factor_g_pCa: 3.0
      factor_g_pK: 0.5
      factor_f_tau: 1.5
    1.8:
      factor_g_Kr: 1.1241830065359477
      factor_g_Ks: 1.125
      factor_g_pCa: 7.0
      factor_g_pK: 0.15
      factor_f_tau: 2.0
  epi:
    g_Ks: 0.392
    g_to: 0.294
    s_offset: 20.0
    s_variant: 0.0
  endo:
    g_Ks: 0.392
    g_to: 0.073
    s_offset: 28.0
    s_variant: 1.0
  m:
    g_Ks: 0.098
    g_to: 0.294
    s_offset: 20.0
    s_variant: 0.0
example discretisation:
  dt: 0.021
  dx: 0.84
  stimulus: 47.5
units:
  t: ms
  x: mm
  u: mV