Luo & Rudy 1991

Key: luo1991model

This model is a significant update of the Beeler-Reuter mammalian ventricular model (1977), and like the Beeler-Reuter model, the Luo-Rudy I model uses Hodgkin-Huxley type equations to calculate ionic currents.

References

  1. https://doi.org/10.1161/01.res.68.6.1501

Figure

Figure illustrating the Luo & Rudy 1991 model.

Variables

  1. V = -84.3801107371

  2. m = 0.00171338077730188

  3. h = 0.982660523699656

  4. j = 0.989108212766685

  5. d = 0.00302126301779861

  6. f = 0.999967936476325

  7. X = 0.0417603108167287

  8. Cai = 0.00017948816388306

Parameters

  • diffusivity_V = 1.0

  • background_current_E_b = -59.87

  • background_current_g_b = 0.03921

  • environment_time = 0.0

  • membrane_C = 1.0

  • membrane_stim_amplitude = -25.5

  • membrane_stim_duration = 2.0

  • membrane_stim_end = 9000.0

  • membrane_stim_period = 1000.0

  • membrane_stim_start = 100.0

  • fast_sodium_current_E_Na = 54.79446393509185

  • fast_sodium_current_g_Na = 23.0

  • slow_inward_current_g_si = 0.09

  • time_dependent_potassium_current_g_K = 0.282

  • time_dependent_potassium_current_E_K = -77.56758438531939

  • time_independent_potassium_current_E_K1 = -87.8929017138025

  • time_independent_potassium_current_g_K1 = 0.6047

  • plateau_potassium_current_E_Kp = -87.8929017138025

  • plateau_potassium_current_g_Kp = 0.0183

  • ionic_concentrations_Ko = 5.4

Source code

OpenCL kernel
// background_current
const Real background_current_i_b = background_current_g_b * (V - background_current_E_b);

// fast_sodium_current_h_gate
const Real fast_sodium_current_h_gate_alpha_h = ((V < -40.0) ? 0.135 * exp((80.0 + V) / -6.8) : 0.0);
const Real fast_sodium_current_h_gate_beta_h = ((V < -40.0) ? 3.56 * exp(0.079 * V) + 310000.0 * exp(0.35 * V) : 1.0 / ((fabs(0.13 * (1.0 + exp((V + 10.66) / -11.1))) < VERY_SMALL_NUMBER) ? ((0.13 * (1.0 + exp((V + 10.66) / -11.1)) < 0.0) ? -VERY_SMALL_NUMBER : VERY_SMALL_NUMBER) : 0.13 * (1.0 + exp((V + 10.66) / -11.1))));
const Real tau_h = 1.0 / (fast_sodium_current_h_gate_alpha_h + fast_sodium_current_h_gate_beta_h);
const Real h_inf = fast_sodium_current_h_gate_alpha_h * tau_h;
*_new_h = h_inf + (h - h_inf) * exp(-dt / tau_h);

// fast_sodium_current_j_gate
const Real fast_sodium_current_j_gate_alpha_j = ((V < -40.0) ? (-127140.0 * exp(0.2444 * V) - 3.474e-05 * exp(-0.04391 * V)) * (V + 37.78) / ((fabs(1.0 + exp(0.311 * (V + 79.23))) < VERY_SMALL_NUMBER) ? ((1.0 + exp(0.311 * (V + 79.23)) < 0.0) ? -VERY_SMALL_NUMBER : VERY_SMALL_NUMBER) : 1.0 + exp(0.311 * (V + 79.23))) : 0.0);
const Real fast_sodium_current_j_gate_beta_j = ((V < -40.0) ? 0.1212 * exp(-0.01052 * V) / ((fabs(1.0 + exp(-0.1378 * (V + 40.14))) < VERY_SMALL_NUMBER) ? ((1.0 + exp(-0.1378 * (V + 40.14)) < 0.0) ? -VERY_SMALL_NUMBER : VERY_SMALL_NUMBER) : 1.0 + exp(-0.1378 * (V + 40.14))) : 0.3 * exp(-2.535e-07 * V) / ((fabs(1.0 + exp(-0.1 * (V + 32.0))) < VERY_SMALL_NUMBER) ? ((1.0 + exp(-0.1 * (V + 32.0)) < 0.0) ? -VERY_SMALL_NUMBER : VERY_SMALL_NUMBER) : 1.0 + exp(-0.1 * (V + 32.0))));
const Real tau_j = 1.0 / (fast_sodium_current_j_gate_alpha_j + fast_sodium_current_j_gate_beta_j);
const Real j_inf = fast_sodium_current_j_gate_alpha_j * tau_j;
*_new_j = j_inf + (j - j_inf) * exp(-dt / tau_j);

// fast_sodium_current_m_gate
const Real fast_sodium_current_m_gate_alpha_m = 0.32 * (V + 47.13) / ((fabs(1.0 - exp(-0.1 * (V + 47.13))) < VERY_SMALL_NUMBER) ? ((1.0 - exp(-0.1 * (V + 47.13)) < 0.0) ? -VERY_SMALL_NUMBER : VERY_SMALL_NUMBER) : 1.0 - exp(-0.1 * (V + 47.13)));
const Real fast_sodium_current_m_gate_beta_m = 0.08 * exp(-V / 11.0);
const Real tau_m = 1.0 / (fast_sodium_current_m_gate_alpha_m + fast_sodium_current_m_gate_beta_m);
const Real m_inf = fast_sodium_current_m_gate_alpha_m * tau_m;
*_new_m = m_inf + (m - m_inf) * exp(-dt / tau_m);

// slow_inward_current
const Real slow_inward_current_E_si = 7.7 - 13.0287 * log(Cai / 1.0);
const Real slow_inward_current_i_si = slow_inward_current_g_si * d * f * (V - slow_inward_current_E_si);

// slow_inward_current_d_gate
const Real slow_inward_current_d_gate_alpha_d = 0.095 * exp(-0.01 * (V - 5.0)) / ((fabs(1.0 + exp(-0.072 * (V - 5.0))) < VERY_SMALL_NUMBER) ? ((1.0 + exp(-0.072 * (V - 5.0)) < 0.0) ? -VERY_SMALL_NUMBER : VERY_SMALL_NUMBER) : 1.0 + exp(-0.072 * (V - 5.0)));
const Real slow_inward_current_d_gate_beta_d = 0.07 * exp(-0.017 * (V + 44.0)) / ((fabs(1.0 + exp(0.05 * (V + 44.0))) < VERY_SMALL_NUMBER) ? ((1.0 + exp(0.05 * (V + 44.0)) < 0.0) ? -VERY_SMALL_NUMBER : VERY_SMALL_NUMBER) : 1.0 + exp(0.05 * (V + 44.0)));
const Real tau_d = 1.0 / (slow_inward_current_d_gate_alpha_d + slow_inward_current_d_gate_beta_d);
const Real d_inf = slow_inward_current_d_gate_alpha_d * tau_d;
*_new_d = d_inf + (d - d_inf) * exp(-dt / tau_d);

// slow_inward_current_f_gate
const Real slow_inward_current_f_gate_alpha_f = 0.012 * exp(-0.008 * (V + 28.0)) / ((fabs(1.0 + exp(0.15 * (V + 28.0))) < VERY_SMALL_NUMBER) ? ((1.0 + exp(0.15 * (V + 28.0)) < 0.0) ? -VERY_SMALL_NUMBER : VERY_SMALL_NUMBER) : 1.0 + exp(0.15 * (V + 28.0)));
const Real slow_inward_current_f_gate_beta_f = 0.0065 * exp(-0.02 * (V + 30.0)) / ((fabs(1.0 + exp(-0.2 * (V + 30.0))) < VERY_SMALL_NUMBER) ? ((1.0 + exp(-0.2 * (V + 30.0)) < 0.0) ? -VERY_SMALL_NUMBER : VERY_SMALL_NUMBER) : 1.0 + exp(-0.2 * (V + 30.0)));
const Real tau_f = 1.0 / (slow_inward_current_f_gate_alpha_f + slow_inward_current_f_gate_beta_f);
const Real f_inf = slow_inward_current_f_gate_alpha_f * tau_f;
*_new_f = f_inf + (f - f_inf) * exp(-dt / tau_f);

// time_dependent_potassium_current_X_gate
const Real time_dependent_potassium_current_X_gate_alpha_X = 0.0005 * exp(0.083 * (V + 50.0)) / ((fabs(1.0 + exp(0.057 * (V + 50.0))) < VERY_SMALL_NUMBER) ? ((1.0 + exp(0.057 * (V + 50.0)) < 0.0) ? -VERY_SMALL_NUMBER : VERY_SMALL_NUMBER) : 1.0 + exp(0.057 * (V + 50.0)));
const Real time_dependent_potassium_current_X_gate_beta_X = 0.0013 * exp(-0.06 * (V + 20.0)) / ((fabs(1.0 + exp(-0.04 * (V + 20.0))) < VERY_SMALL_NUMBER) ? ((1.0 + exp(-0.04 * (V + 20.0)) < 0.0) ? -VERY_SMALL_NUMBER : VERY_SMALL_NUMBER) : 1.0 + exp(-0.04 * (V + 20.0)));
const Real tau_X = safe_divide(1.0, time_dependent_potassium_current_X_gate_alpha_X + time_dependent_potassium_current_X_gate_beta_X);
const Real X_inf = time_dependent_potassium_current_X_gate_alpha_X * tau_X;
*_new_X = X_inf + (X - X_inf) * exp(-safe_divide(dt, tau_X));

// time_dependent_potassium_current_Xi_gate
const Real time_dependent_potassium_current_Xi_gate_Xi = ((V > -100.0) ? 2.837 * (exp(0.04 * (V + 77.0)) - 1.0) / ((fabs((V + 77.0) * exp(0.04 * (V + 35.0))) < VERY_SMALL_NUMBER) ? (((V + 77.0) * exp(0.04 * (V + 35.0)) < 0.0) ? -VERY_SMALL_NUMBER : VERY_SMALL_NUMBER) : (V + 77.0) * exp(0.04 * (V + 35.0))) : 1.0);

// intracellular_calcium_concentration
*_new_Cai = Cai + dt*(-0.0001 / 1.0 * slow_inward_current_i_si + 0.07 * (0.0001 - Cai));

// membrane
const Real membrane_I_stim = ((((environment_time >= membrane_stim_start) && (environment_time <= membrane_stim_end)) && (environment_time - membrane_stim_start - floor((environment_time - membrane_stim_start) / membrane_stim_period) * membrane_stim_period <= membrane_stim_duration)) ? membrane_stim_amplitude : 0.0);

// fast_sodium_current
const Real fast_sodium_current_i_Na = fast_sodium_current_g_Na * pow(m, 3.0) * h * j * (V - fast_sodium_current_E_Na);

// time_dependent_potassium_current
const Real time_dependent_potassium_current_i_K = time_dependent_potassium_current_g_K * sqrt(ionic_concentrations_Ko/5.4) * X * time_dependent_potassium_current_Xi_gate_Xi * (V - time_dependent_potassium_current_E_K);

// time_independent_potassium_current_K1_gate
const Real time_independent_potassium_current_K1_gate_alpha_K1 = 1.02 / ((fabs(1.0 + exp(0.2385 * (V - time_independent_potassium_current_E_K1 - 59.215))) < VERY_SMALL_NUMBER) ? ((1.0 + exp(0.2385 * (V - time_independent_potassium_current_E_K1 - 59.215)) < 0.0) ? -VERY_SMALL_NUMBER : VERY_SMALL_NUMBER) : 1.0 + exp(0.2385 * (V - time_independent_potassium_current_E_K1 - 59.215)));
const Real time_independent_potassium_current_K1_gate_beta_K1 = (0.49124 * exp(0.08032 * (V + 5.476 - time_independent_potassium_current_E_K1)) + 1.0 * exp(0.06175 * (V - (time_independent_potassium_current_E_K1 + 594.31)))) / ((fabs(1.0 + exp(-0.5143 * (V - time_independent_potassium_current_E_K1 + 4.753))) < VERY_SMALL_NUMBER) ? ((1.0 + exp(-0.5143 * (V - time_independent_potassium_current_E_K1 + 4.753)) < 0.0) ? -VERY_SMALL_NUMBER : VERY_SMALL_NUMBER) : 1.0 + exp(-0.5143 * (V - time_independent_potassium_current_E_K1 + 4.753)));
const Real time_independent_potassium_current_K1_gate_K1_infinity = time_independent_potassium_current_K1_gate_alpha_K1 / ((fabs(time_independent_potassium_current_K1_gate_alpha_K1 + time_independent_potassium_current_K1_gate_beta_K1) < VERY_SMALL_NUMBER) ? ((time_independent_potassium_current_K1_gate_alpha_K1 + time_independent_potassium_current_K1_gate_beta_K1 < 0.0) ? -VERY_SMALL_NUMBER : VERY_SMALL_NUMBER) : time_independent_potassium_current_K1_gate_alpha_K1 + time_independent_potassium_current_K1_gate_beta_K1);

// plateau_potassium_current
const Real plateau_potassium_current_Kp = 1.0 / ((fabs(1.0 + exp((7.488 - V) / 5.98)) < VERY_SMALL_NUMBER) ? ((1.0 + exp((7.488 - V) / 5.98) < 0.0) ? -VERY_SMALL_NUMBER : VERY_SMALL_NUMBER) : 1.0 + exp((7.488 - V) / 5.98));
const Real plateau_potassium_current_i_Kp = plateau_potassium_current_g_Kp * plateau_potassium_current_Kp * (V - plateau_potassium_current_E_Kp);

// *remaining*
const Real time_independent_potassium_current_i_K1 = time_independent_potassium_current_g_K1 * sqrt(ionic_concentrations_Ko/5.4) * time_independent_potassium_current_K1_gate_K1_infinity * (V - time_independent_potassium_current_E_K1);
*_new_V = V + dt*(-1.0 / membrane_C * (membrane_I_stim + fast_sodium_current_i_Na + slow_inward_current_i_si + time_dependent_potassium_current_i_K + time_independent_potassium_current_i_K1 + plateau_potassium_current_i_Kp + background_current_i_b) + _diffuse_V);

Additional metadata

keywords:
- excitable media
- electrophysiology
- heart
- human
- ventricle
extra parameters:
  ionic_concentrations_Ki: 145.0
  ionic_concentrations_Nai: 18.0
  ionic_concentrations_Nao: 140.0
  membrane_F: 96484.6
  membrane_R: 8314.0
  membrane_T: 310.0
  time_dependent_potassium_current_PR_NaK: 0.01833
example discretisation:
  dt: 0.016
  dx: 0.48
  stimulus: 60.4
units:
  t: ms
  x: mm
  u: mV