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tentusscher-2006-modified.mmt
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tentusscher-2006-modified.mmt
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[[model]]
desc: """
A modified version of the 2006 update of the Ten Tusscher et al. model.
This version has:
- Depleted initial SR (from 3.64 to 2.5 mM)
- Reduced ICaL
- Increased epicardial and mid-myocardial Ito
- Voltage shifts in Ito activation and inactivation
- Doubled stimulus amplitude
"""
# Initial values
membrane.V = -85.23
calcium.Cai = 0.000126
calcium.CaSR = 2.5 # 3.64
calcium.CaSS = 0.00036
sodium.Nai = 8.604
potassium.Ki = 136.89
ina.m = 0.00172
ina.h = 0.7444
ina.j = 0.7045
ikr.xr1 = 0.00621
ikr.xr2 = 0.4712
iks.xs = 0.0095
ito.r = 2.42e-8
ito.s = 0.999998
ical.d = 3.373e-5
ical.f = 0.7888
ical.f2 = 0.9755
ical.fCaSS = 0.9953
jrel.R_prime = 0.9073
#
# Simulator variables
#
[engine]
time = 0 [ms]
in [ms]
bind time
pace = 0
bind pace
#
# Membrane potential
#
# Appendix to [3]
#
[membrane]
use stimulus.i_stim
dot(V) = -(i_ion + i_stim + i_diff)
in [mV]
label membrane_potential
i_ion = (+ ina.INa
+ ik1.IK1
+ ikr.IKr
+ iks.IKs
+ ito.Ito
+ ical.ICaL
+ inak.INaK
+ inaca.INaCa
+ ipca.IpCa
+ ipk.IpK
+ icab.ICab
+ inab.INab
)
in [A/F]
i_diff = 0 [A/F]
in [A/F]
bind diffusion_current
#
# Stimulus current
#
# Appendix to [3]
#
[stimulus]
i_stim = engine.pace * amplitude
in [A/F]
amplitude = 2 * -52 [A/F]
in [A/F]
#
# Cell parameters
#
[cell]
type = 1
desc: The type of cell: endocardial = 0, epicardial = 1, mid-myocardial = 2
Vc = 16404 [um^3]
in [um^3]
desc: Bulk cytoplasm volume
Vss = 54.68 [um^3]
in [um^3]
desc: Dyadic (junctional) subspace volume
Vsr = 1094 [um^3]
in [um^3]
desc: Sarcoplasmic reticulum volume
Cm = 185 [pF]
in [pF]
desc: Cell capacitance
#
# Physical constants
#
# Appendix to [3]
#
[phys]
F = 96.485 [C/mmol]
in [C/mmol]
R = 8.314 [J/mol/K]
in [J/mol/K]
T = 310 [K]
in [K]
RTF = R * T / F
in [mV]
FRT = F / (R * T)
in [1/mV]
FFRT = F * FRT
in [C/mmol/mV]
#
# Fast sodium current
#
# Appendix to [3]
#
[ina]
use membrane.V, nernst.ENa
INa = gNa * m ^ 3 * h * j * (V - ENa)
in [A/F]
gNa = 14.838 [mS/uF]
in [mS/uF]
dot(m) = (inf - m) / tau
inf = 1 / (1 + exp((-56.86 [mV] - V) / 9.03 [mV])) ^ 2
tau = 1 [ms] * alpha * beta
in [ms]
alpha = 1 / (1 + exp((-60 [mV] - V) / 5 [mV]))
beta = 0.1 / (1 + exp((V + 35 [mV]) / 5 [mV])) + 0.1 / (1 + exp((V - 50 [mV]) / 200 [mV]))
dot(h) = (inf - h) / tau
inf = 1 / (1 + exp((V + 71.55 [mV]) / 7.43 [mV])) ^ 2
tau = 1 / (alpha + beta)
in [ms]
alpha = if(V < -40 [mV],
0.057 [mS/uF] * exp(-(V + 80 [mV]) / 6.8 [mV]),
0 [mS/uF])
in [1/ms]
beta = if(V < -40 [mV],
2.7 [mS/uF] * exp(0.079 [1/mV] * V) + 310000 [mS/uF] * exp(0.3485 [1/mV] * V),
0.77 [mS/uF] / (0.13 * (1 + exp((V + 10.66 [mV]) / -11.1 [mV]))))
in [1/ms]
dot(j) = (inf - j) / tau
inf = 1 / (1 + exp((V + 71.55 [mV]) / 7.43 [mV])) ^ 2
tau = 1 / (alpha + beta)
in [ms]
alpha = if(V < -40 [mV],
(-25428 [mS/uF] * exp(0.2444 [1/mV] * V) - 6.948e-6 [mS/uF] * exp(-0.04391 [1/mV] * V)) * (V + 37.78 [mV]) / 1 [mV] / (1 + exp(0.311 [1/mV] * (V + 79.23 [mV]))),
0 [mS/uF])
in [1/ms]
beta = if(V < -40 [mV],
0.02424 [mS/uF] * exp(-0.01052 [1/mV] * V) / (1 + exp(-0.1378 [1/mV] * (V + 40.14 [mV]))),
0.6 [mS/uF] * exp(0.057 [1/mV] * V) / (1 + exp(-0.1 [1/mV] * (V + 32 [mV]))))
in [1/ms]
#
# Inward rectifier potassium current
#
# Appendix to [3]
#
[ik1]
use membrane.V, nernst.EK
IK1 = gK1 * inf * (V - EK)
in [A/F]
gK1 = 5.405 [mS/uF] * sqrt(extra.Ko / 5.4 [mM])
in [mS/uF]
inf = alpha / (alpha + beta)
alpha = 0.1 [1/ms] / (1 + exp(0.06 [1/mV] * (V - EK - 200 [mV])))
in [1/ms]
beta = (3 [1/ms] * exp(0.0002 [1/mV] * (V - EK + 100 [mV])) + 1 [1/ms] * exp(0.1 [1/mV] * (V - EK - 10 [mV]))) / (1 + exp(-0.5 [1/mV] * (V - EK)))
in [1/ms]
#
# Rapid time-dependent potassium current
#
# Appendix to [3]
#
[ikr]
use membrane.V, nernst.EK
IKr = gKr * xr1 * xr2 * (V - EK)
in [A/F]
gKr = 0.153 [mS/uF] * sqrt(extra.Ko / 5.4 [mM])
in [mS/uF]
dot(xr1) = (inf - xr1) / tau
inf = 1 / (1 + exp((-26 [mV] - V) / 7 [mV]))
tau = 1 [ms] * alpha * beta
in [ms]
alpha = 450 / (1 + exp((-45 [mV] - V) / 10 [mV]))
beta = 6 / (1 + exp((V + 30 [mV]) / 11.5 [mV]))
dot(xr2) = (inf - xr2) / tau
inf = 1 / (1 + exp((V + 88 [mV]) / 24 [mV]))
tau = 1 [ms] * alpha * beta
in [ms]
alpha = 3 / (1 + exp((-60 [mV] - V) / 20 [mV]))
beta = 1.12 / (1 + exp((V - 60 [mV]) / 20 [mV]))
#
# Slow time-dependent potassium current
#
# Appendix to [2]
#
[iks]
use membrane.V, nernst.EKs
IKs = gKs * xs ^ 2 * (V - EKs)
in [A/F]
gKs = if(cell.type != 2,
0.392 [mS/uF], # Endo and epicardial
0.098 [mS/uF] # Mid-myocardial
)
in [mS/uF]
dot(xs) = (inf - xs) / tau
inf = 1 / (1 + exp((-5 [mV] - V) / 14 [mV]))
tau = 1 [ms] * alpha * beta + 80 [ms]
in [ms]
alpha = 1400 / sqrt(1 + exp((5 [mV] - V) / 6 [mV]))
beta = 1 / (1 + exp((V - 35 [mV]) / 15 [mV]))
#
# Transient outward current
#
# Appendix to [2]
#
[ito]
use nernst.EK
Ito = gto * r * s * (membrane.V - EK)
in [A/F]
gto = if(cell.type == 0,
0.073 [mS/uF], # Endocardial
#0.294 [mS/uF] # Epicardial and mid-myocardial
0.7 [mS/uF]
)
in [mS/uF]
dot(r) = (inf - r) / tau
v20 = membrane.V + 20 [mV]
in [mV]
inf = 1 / (1 + exp((20 [mV] - v20) / 6 [mV]))
tau = 9.5 [ms] * exp(-(v20 + 40 [mV]) ^ 2 / 1800 [mV^2]) + 0.8 [ms]
in [ms]
dot(s) = (inf - s) / tau
v10 = membrane.V + 10 [mV]
in [mV]
inf = if(cell.type == 0,
1 / (1 + exp((v10 + 28 [mV]) / 5 [mV])), # Endocardial
1 / (1 + exp((v10 + 20 [mV]) / 5 [mV])) # Epicardial and mid-myocardial
)
tau = if(cell.type == 0,
# Endocardial:
1000 [ms] * exp(-(v10 + 67 [mV]) ^ 2 / 1000 [mV^2]) + 8 [ms],
# Epicardial and mid-myocardial:
85 [ms] * exp(-(v10 + 45 [mV]) ^ 2 / 320 [mV^2]) + 5 [ms] / (1 + exp((v10 - 20 [mV]) / 5 [mV])) + 3 [ms]
)
in [ms]
#
# L-type calcium current
#
# Appendix to [2]
#
[ical]
use membrane.V
use phys.FRT, phys.FFRT
use extra.Cao, calcium.Cai, calcium.CaSS
ICaL = gCaL * d * f * f2 * fCaSS * 4 * (V - 15 [mV]) * FFRT * (0.25 * CaSS * exp(2 * (V - 15 [mV]) * FRT) - Cao) / (exp(2 * (V - 15 [mV]) * FRT) - 1)
in [A/F]
# Reduced ICaL
#gCaL = 0.0398 [L/F/s]
gCaL = 0.02 [L/F/s]
in [L/F/s]
dot(d) = (inf - d) / tau
inf = 1 / (1 + exp((-8 [mV] - V) / 7.5 [mV]))
tau = 1 [ms] * (alpha * beta + gamma)
in [ms]
alpha = 1.4 / (1 + exp((-35 [mV] - V) / 13 [mV])) + 0.25
beta = 1.4 / (1 + exp((V + 5 [mV]) / 5 [mV]))
gamma = 1 / (1 + exp((50 [mV] - V) / 20 [mV]))
dot(f) = (inf - f) / tau
inf = 1 / (1 + exp((V + 20 [mV]) / 7 [mV]))
tau = 1102.5 [ms] * exp(-(V + 27 [mV]) ^ 2 / 225 [mV^2]) + 200 [ms] / (1 + exp((13 [mV] - V) / 10 [mV])) + 180 [ms] / (1 + exp((V + 30 [mV]) / 10 [mV])) + 20 [ms]
in [ms]
dot(f2) = (inf - f2) / tau
inf = 0.67 / (1 + exp((V + 35 [mV]) / 7 [mV])) + 0.33
tau = 562 [ms] * exp(-(V + 27 [mV]) ^ 2 / 240 [mV^2]) + 31 [ms] / (1 + exp((25 [mV] - V) / 10 [mV])) + 80 [ms] / (1 + exp((V + 30 [mV]) / 10 [mV]))
in [ms]
dot(fCaSS) = (inf - fCaSS) / tau
inf = 0.6 / (1 + (CaSS / 0.05 [mM]) ^ 2) + 0.4
tau = 80 [ms] / (1 + (CaSS / 0.05 [mM]) ^ 2) + 2 [ms]
in [ms]
#
# Sodium-potassium pump
#
# Appendix to [3]
#
[inak]
use membrane.V, phys.FRT
use extra.Ko, sodium.Nai
INaK = P_NaK * Ko / (Ko + K_mk) * Nai / (Nai + K_mNa) / (1 + 0.1245 * exp(-0.1 * V * FRT) + 0.0353 * exp(-V * FRT))
in [A/F]
P_NaK = 2.724 [A/F]
in [A/F]
K_mNa = 40 [mM]
in [mM]
K_mk = 1 [mM]
in [mM]
#
# Sodium-calcium exchanger
#
# Appendix to [3]
#
[inaca]
use membrane.V, phys.FRT
use extra.Nao, extra.Cao
use sodium.Nai, calcium.Cai
INaCa = (K_NaCa
* (exp(gamma * V * FRT) * Nai ^ 3 * Cao - exp((gamma - 1) * V * FRT) * Nao ^ 3 * Cai * alpha)
/ ((Km_Nai ^ 3 + Nao ^ 3) * (Km_Ca + Cao) * (1 + K_sat * exp((gamma - 1) * V * FRT))))
in [A/F]
K_NaCa = 1000 [A/F]
in [A/F]
Km_Ca = 1.38 [mM]
in [mM]
Km_Nai = 87.5 [mM]
in [mM]
K_sat = 0.1
alpha = 2.5
gamma = 0.35
#
# Calcium pump
#
# Appendix to [3]
#
[ipca]
use calcium.Cai
IpCa = gpCa * Cai / (Cai + KpCa)
in [A/F]
gpCa = 0.1238 [A/F]
in [A/F]
KpCa = 0.0005 [mM]
in [mM]
#
# Potassium pump
#
# Appendix to [3]
#
[ipk]
use membrane.V, nernst.EK
IpK = gpK * (V - EK) / (1 + exp((25 [mV] - V) / 5.98 [mV]))
in [A/F]
gpK = 0.0146 [mS/uF]
in [mS/uF]
#
# Background calcium current
#
# Appendix to [3]
#
[icab]
use membrane.V, nernst.ECa
ICab = gCab * (V - ECa)
in [A/F]
gCab = 0.000592 [mS/uF]
in [mS/uF]
#
# Background sodium current
#
# Appendix to [3]
#
[inab]
use membrane.V, nernst.ENa
INab = gNab * (V - ENa)
in [A/F]
gNab = 0.00029 [mS/uF]
in [mS/uF]
#
# External concentrations
#
# Appendix to [3]
#
[extra]
Cao = 2 [mM]
in [mM]
Nao = 140 [mM]
in [mM]
Ko = 5.4 [mM]
in [mM]
#
# Nernst/Reversal potentials
#
# Appendix to [3]
#
[nernst]
use extra.Cao, calcium.Cai
use extra.Nao, sodium.Nai
use extra.Ko, potassium.Ki
use phys.RTF
ECa = RTF * log(Cao / Cai) * 0.5
in [mV]
ENa = RTF * log(Nao / Nai)
in [mV]
EK = RTF * log(Ko / Ki)
in [mV]
EKs = RTF * log((Ko + P_kna * Nao) / (Ki + P_kna * Nai))
in [mV]
P_kna = 0.03
#
# Calcium release from the SR (RyR)
#
[jrel]
use membrane.V
use calcium.CaSS, calcium.CaSR
Jrel = Vrel * O * (CaSR - CaSS)
in [mM/ms]
Vrel = 0.102 [mS/uF]
in [mS/uF]
O = k1 * CaSS ^ 2 * R_prime / (k3 + k1 * CaSS ^ 2)
kcasr = max_sr - (max_sr - min_sr) / (1 + (EC / CaSR) ^ 2)
max_sr = 2.5
min_sr = 1
EC = 1.5 [mM]
in [mM]
k1 = k1_prime / kcasr
in [1/mM^2/ms]
k1_prime = 0.15 [1/mM^2/ms]
in [1/mM^2/ms]
k2 = k2_prime * kcasr
in [1/mM/ms]
k2_prime = 0.045 [1/mM/ms]
in [1/mM/ms]
k3 = 0.06 [mS/uF]
in [mS/uF]
k4 = 0.005 [mS/uF]
in [mS/uF]
dot(R_prime) = -k2 * CaSS * R_prime + k4 * (1 - R_prime)
#
# Leak from the SR
#
[jleak]
use calcium.Cai, calcium.CaSR
Jleak = Vleak * (CaSR - Cai)
in [mM/ms]
Vleak = 0.00036 [mS/uF]
in [mS/uF]
#
# Calcium uptake into the SR (SERCA)
#
[jup]
use calcium.Cai
Jup = Vmax_up / (1 + K_up ^ 2 / Cai ^ 2)
in [mM/ms]
Vmax_up = 0.006375 [mM/ms]
in [mM/ms]
K_up = 0.00025 [mM]
in [mM]
#
# Diffusion from SS to Bulk
#
[jxfer]
use calcium.Cai, calcium.CaSS
Jxfer = Vxfer * (CaSS - Cai)
in [mM/ms]
Vxfer = 0.0038 [mS/uF]
in [mS/uF]
#
# Calcium dynamics
#
# Appendix to [2]
#
[calcium]
use jup.Jup, jrel.Jrel, jleak.Jleak, jxfer.Jxfer
use phys.F, cell.Cm, cell.Vc, cell.Vss, cell.Vsr
# Free calcium in cytosol, SS, and SR
dot(Cai) = ddt_Cai_total * f_JCai_free
in [mM]
dot(CaSS) = ddt_CaSS_total * f_JCaSS_free
in [mM]
dot(CaSR) = ddt_CaSR_total * f_JCaSR_free
in [mM]
# Derivative of total calcium concentrations (free + buffered)
ddt_Cai_total = -(icab.ICab + ipca.IpCa - 2 * inaca.INaCa) * Cm / (2 * Vc * F) + (Jleak - Jup) * Vsr / Vc + Jxfer
in [mM/ms]
ddt_CaSS_total = -ical.ICaL * Cm / (2 * Vss * F) + Jrel * Vsr / Vss - Jxfer * Vc / Vss
in [mM/ms]
ddt_CaSR_total = Jup - (Jrel + Jleak)
in [mM/ms]
# Conversion factors from d/dt total to d/dt free
f_JCai_free = 1 / (1 + Buf_c * K_buf_c / (Cai + K_buf_c) ^ 2)
f_JCaSS_free = 1 / (1 + Buf_SS * K_buf_SS / (CaSS + K_buf_SS) ^ 2)
f_JCaSR_free = 1 / (1 + Buf_SR * K_buf_SR / (CaSR + K_buf_SR) ^ 2)
# Buffering: Ca_buffered = Ca_i * Buf_c / (Ca_i + K_bufc)
Buf_c = 0.2 [mM]
in [mM]
Buf_SS = 0.4 [mM]
in [mM]
Buf_SR = 10 [mM]
in [mM]
K_buf_c = 0.001 [mM]
in [mM]
K_buf_SS = 0.00025 [mM]
in [mM]
K_buf_SR = 0.3 [mM]
in [mM]
#
# Sodium dynamics
#
# Appendix to [3]
#
[sodium]
use phys.F, cell.Cm, cell.Vc
dot(Nai) = -INa_total * Cm / (Vc * F)
in [mM]
INa_total = ina.INa + inab.INab + 3 * inak.INaK + 3 * inaca.INaCa
in [A/F]
#
# Potassium dynamics
#
# Appendix to [3]
#
[potassium]
use phys.F, cell.Cm, cell.Vc
dot(Ki) = -IK_total * Cm / (Vc * F)
in [mM]
IK_total = ik1.IK1 + ito.Ito + ikr.IKr + iks.IKs + ipk.IpK + stimulus.i_stim - 2 * inak.INaK
in [A/F]
[[protocol]]
# Level Start Length Period Multiplier
1 2 3 400 0