Model Mathematics

vol_cyt = fr_cyt vol_pmu vol_pmu = \frac{\frac{π {dia}^{3}}{6} 1 e 9}{1 1 1} V_tau = \frac{R T}{F}

SVR_pmu = \frac{svr}{dia} C_m = C_sp A_pmu VD = \frac{V}{V_tau} V = \frac{F vol_cyt}{C_sp A_pmu} (((K_i - K_o) + 2 (Ca_i - Ca_o) + Na_i - Na_o) + anoff) anoff = \frac{(- 52.31) C_sp A_pmu}{F vol_cyt} - (2 (1.5 e -4 - Ca_o)) - (6 - Na_o) - (140 - K_o) A_pmu = \frac{SVR_pmu vol_pmu 1 e -3 1 e -3 1 e -3}{1} V_Ca = 0.5 \ln (\frac{Ca_o}{Ca_i}) V_Na = \ln (\frac{Na_o}{Na_i}) V_K = \ln (\frac{K_o}{K_i})

\frac{d}{d t} (Ca_i) = J_Ca \frac{d}{d t} (Na_i) = J_Na \frac{d}{d t} (K_i) = J_K

J_Ca = J_ca - (J_calb + 4 J_cam) J_ca = \frac{- 1}{2 F vol_cyt} (I_CaL + I_CaT + I_CaHVA + 2 I_pmca - (2 I_xm))

J_Na = \frac{- 1}{F vol_cyt} (3 I_nk + 3 I_xm + I_Na + I_Nalk + I_NaHCN)

I_K = I_Ksk + I_KHCN + I_Ka + I_Kdr + I_Kir J_K = \frac{- 1}{F vol_cyt} (I_K - (2 I_nk))

J_calb = kcal_1 Calb Ca_i - (kcal_2 CaCalb) J_cam = alpha_cam Cam - (beta_cam CaCam) kcam_cb = 12000 {Ca_i}^{2} kcam_nb = 3.7 e 6 {Ca_i}^{2} alpha_cam = kcam_cb kcam_nb (\frac{1}{kcam_cb + kcam_nd} + \frac{1}{kcam_cd + kcam_nd}) beta_cam = kcam_cd kcam_nd (\frac{1}{kcam_cb + kcam_nd} + \frac{1}{kcam_cd + kcam_nd}) CaCalb = Calbtot - Calb CaCam = Camtot - Cam \frac{d}{d t} (Calb) = - J_calb \frac{d}{d t} (Cam) = - J_cam

I_CaL = \frac{g_cal m_cal h_cal \sqrt{Ca_i Ca_o} \sinh (VD - V_Ca)}{\frac{\sinh VD}{VD}} \frac{d}{d t} (m_cal) = \frac{\frac{1}{1 + ⅇ^{\frac{- (V + 15)}{7}}} - m_cal}{10 ⅇ^{- ({(\frac{V + 86.4}{23.2})}^{2})} + 0.943} h_cal = \frac{0.00045}{0.00045 + Ca_i}

I_CaT = \frac{g_cat m_cat h_cat \sqrt{Ca_i Ca_o} \sinh (VD - V_Ca)}{\frac{\sinh VD}{VD}} \frac{d}{d t} (m_cat) = \frac{\frac{1}{1 + ⅇ^{\frac{- (V + 63)}{1.5}}} - m_cat}{65 ⅇ^{- ({(\frac{V + 68}{6})}^{2})} + 12} \frac{d}{d t} (h_cat) = \frac{\frac{1}{1 + ⅇ^{\frac{V + 76.2}{3}}} - h_cat}{50 ⅇ^{- ({(\frac{V + 72}{10})}^{2})} + 10}

I_CaHVA = \frac{g_cah m_cah h_cah \sqrt{Ca_i Ca_o} \sinh (VD - V_Ca)}{\frac{\sinh VD}{VD}} \frac{d}{d t} (m_cah) = \frac{\frac{1}{1 + ⅇ^{\frac{- (V + 10)}{10}}} - m_cah}{0.1 ⅇ^{- ({(\frac{V + 62}{13})}^{2})} + 0.05} \frac{d}{d t} (h_cah) = \frac{\frac{1}{1 + ⅇ^{\frac{V + 48}{5}}} - h_cah}{0.5 ⅇ^{- ({(\frac{V + 55.6}{18})}^{2})} + 0.5}

O_na = {m_na}^{3} h_na \frac{d}{d t} (m_na) = A_mna ⅇ^{za_mna VD} (1 - m_na) - (B_mna ⅇ^{(- zb_mna) VD} m_na) \frac{d}{d t} (h_na) = A_hna ⅇ^{za_hna VD} (1 - h_na) - (B_hna ⅇ^{(- zb_hna) VD} h_na) I_Na = \frac{g_na O_na \sqrt{Na_i Na_o} \sinh (0.5 (VD - V_Na))}{\frac{\sinh (0.5 VD)}{0.5 VD}}

I_Nalk = \frac{g_nalk \sqrt{Na_i Na_o} \sinh (0.5 (VD - V_Na))}{\frac{\sinh (0.5 VD)}{0.5 VD}}

\frac{d}{d t} (O_hcn) = kf_hcn (1 - O_hcn) - (kr_hcn O_hcn) kf_hcn = kf_free P_c + kf_bnd (1 - P_c) kr_hcn = kr_free P_o + kr_bnd (1 - P_o) kf_free = \frac{0.006}{1 + ⅇ^{\frac{V + 87.7}{6.45}}} kr_free = \frac{0.08}{1 + ⅇ^{\frac{- (V + 51.7)}{7}}} kf_bnd = \frac{0.0268}{1 + ⅇ^{\frac{V + 94.2}{13.3}}} kr_bnd = \frac{0.08}{1 + ⅇ^{\frac{- (V + 35.5)}{7}}} P_c = \frac{1}{1 + \frac{cAMP}{1.163 e -3}} P_o = \frac{1}{1 + \frac{cAMP}{1.45 e -5}} I_NaHCN = \frac{g_nahcn O_hcn \sqrt{Na_i Na_o} \sinh (0.5 (VD - V_Na))}{\frac{\sinh (0.5 VD)}{0.5 VD}} I_KHCN = \frac{g_khcn O_hcn \sqrt{K_i K_o} \sinh (0.5 (VD - V_K))}{\frac{\sinh (0.5 VD)}{0.5 VD}}

O_sk = \frac{{Ca_i}^{4.2}}{{0.35 e -3}^{4.2} + {Ca_i}^{4.2}} I_Ksk = \frac{g_ksk O_sk \sqrt{K_i K_o} \sinh (0.5 (VD - V_K))}{\frac{\sinh (0.5 VD)}{0.5 VD}}

I_Kdr = g_kdr O_kdr (V - (V_K V_tau)) O_kdr = {m_kdr}^{3} \frac{d}{d t} (m_kdr) = \frac{\frac{1}{1 + ⅇ^{\frac{- (V + 25)}{12}}} - m_kdr}{\frac{18}{1 + ⅇ^{\frac{V + 39}{8}}} + 1}

I_Ka = g_ka O_ka (V - (V_K V_tau)) O_ka = {p_ka}^{3} q_ka \frac{d}{d t} (p_ka) = \frac{\frac{1}{1 + ⅇ^{\frac{- (V + 43)}{24}}} - p_ka}{2 ⅇ^{- ({(\frac{V + 50}{23.45})}^{2})} + 1.1} \frac{d}{d t} (q_ka) = \frac{\frac{1}{1 + ⅇ^{\frac{V + 56}{8}}} - q_ka}{20}

I_Kir = g_kir O_kir (V - (V_K V_tau)) O_kir = \frac{1}{1 + ⅇ^{\frac{V + 90}{12.1}}}

I_nk = k_nk (k_1nk P_E1Snk y_nk - (k_2nk P_E2Snk (1 - y_nk))) 1 k_1nk = \frac{0.37}{1 + \frac{0.094}{atp}} Na_eff = Na_o ⅇ^{(- 0.82) VD} \frac{d}{d t} (y_nk) = beta_nk (1 - y_nk) - (alpha_nk y_nk) beta_nk = k_2nk P_E2Snk + k_4nk P_E2Dnk alpha_nk = k_1nk P_E1Snk + k_3nk P_E1Dnk P_E1Snk = \frac{1}{1 + \frac{K_nknai}{Na_i} (1 + \frac{K_i}{K_nkki})} P_E1Dnk = \frac{1}{1 + \frac{K_nkki}{K_i} (1 + \frac{Na_i}{K_nknai})} P_E2Snk = \frac{1}{1 + \frac{K_nknao}{Na_eff} (1 + \frac{K_o}{K_nkko})} P_E2Dnk = \frac{1}{1 + \frac{K_nkko}{K_o} (1 + \frac{Na_eff}{K_nknao})}

I_pmca = K_pmca (k_1pc P_E1Spc y_pc - (k_2pc P_E2Spc (1 - y_pc))) 1 K_pmca = k_pmca (\frac{10.56 CaCam}{CaCam + 0.00005} + 1.2) \frac{d}{d t} (y_pc) = beta_pc (1 - y_pc) - (alpha_pc y_pc) k_1pc = \frac{1}{1 + \frac{0.1}{atp}} P_E1Spc = \frac{1}{1 + \frac{K_pci}{Ca_i}} P_E1pc = 1 - P_E1Spc P_E2Spc = \frac{1}{1 + \frac{K_pco}{Ca_o}} P_E2pc = 1 - P_E2Spc beta_pc = k_2pc P_E2Spc + k_4pc P_E2pc alpha_pc = k_1pc P_E1Spc + k_3pc P_E1pc K_pci = (\frac{173.6}{1 + \frac{CaCam}{0.00005}} + 6.4) 1 e -5

I_xm = k_xm (k_3xm P_E1Dxm y_xm - (k_4xm P_E2Dxm (1 - y_xm))) \frac{d}{d t} (y_xm) = beta_xm (1 - y_xm) - (alpha_xm y_xm) beta_xm = k_2xm P_E2Sxm + k_4xm P_E2Dxm alpha_xm = k_1xm P_E1Sxm + k_3xm P_E1Dxm P_E1Sxm = \frac{1}{1 + \frac{K_xmcai}{Ca_i} (1 + {(\frac{Na_i}{K_xmnai})}^{3})} P_E1Dxm = \frac{1}{1 + {(\frac{K_xmnai}{Na_i})}^{3} (1 + \frac{Ca_i}{K_xmcai})} P_E2Sxm = \frac{1}{1 + \frac{K_xmcao}{Ca_o} (1 + {(\frac{Na_o}{K_xmnao})}^{3})} P_E2Dxm = \frac{1}{1 + {(\frac{K_xmnao}{Na_o})}^{3} (1 + \frac{Ca_o}{K_xmcao})} k_1xm = 1 ⅇ^{(- delta_xmca) VD} k_2xm = 1 ⅇ^{(1 - delta_xmca) VD} k_3xm = 1 ⅇ^{(1 - delta_xmna) VD} k_4xm = 1 ⅇ^{(- delta_xmna) VD}