You are here: Home / Exposures / Katsnelson, Nikitina, Chemla, Solovyova, Coirault, Lecarpentier, Markhasin, 2004 / Influence of viscosity on myocardium mechanical activity: a mathematical model

Model Mathematics

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Component: environment

Component: parameters

Component: parameters_izakov_et_al_1991

q_v = \{\begin{matrix} q_1 - (\frac{q_2 v}{v_max}) if v ≦ 0 \\ q_3 otherwise \end{matrix} v_1 = \frac{v_max}{10}

Component: force

F_CE = lambda p_v N F_muscle = F_XSE F_SE = beta_1 (ⅇ^{alpha_1 (l_2 - l_1)} - 1) F_PE = beta_2 (ⅇ^{alpha_2 l_2} - 1) F_XSE = beta_3 (ⅇ^{alpha_3 l_3} - 1)

Component: crossbridge_kinetics

M_A = \frac{A^{mu}}{A^{mu} + {A_half}^{mu}} n_1 = 0.6 l_1 + 0.5 L_oz = \frac{l_1 + S_0}{0.46 + S_0} k_p_v = chi chi_0 q_v m_0 G_star k_m_v = chi_0 q_v (1 - (chi m_0 G_star)) K_chi = k_p_v M_A n_1 L_oz (1 - N) - (k_m_v N) \frac{d}{d time} (N) = K_chi

Component: length

dl_1_dt = v \frac{d}{d time} (l_1) = dl_1_dt dl_2_dt = w \frac{d}{d time} (l_2) = dl_2_dt dl_3_dt = \{\begin{matrix} 0 if isotonic_mode = 1 \\ - w otherwise \end{matrix} \frac{d}{d time} (l_3) = dl_3_dt

Component: CE_velocity

k_P_vis = \{\begin{matrix} beta_P_lengthening ⅇ^{alpha_P_lengthening l_1} if dl_1_dt ≦ 0 \\ beta_P_shortening ⅇ^{alpha_P_shortening l_1} otherwise \end{matrix} phi_chi = \{\begin{matrix} \frac{lambda K_chi p_v 1 + alpha_P k_P_vis v^{2} + alpha_2 beta_2 ⅇ^{alpha_2 l_2} w 1}{lambda N p_prime_v 1 + k_P_vis} if isotonic_mode = 1 \\ \frac{lambda K_chi p_v 1 + alpha_P k_P_vis v^{2} + (alpha_2 beta_2 ⅇ^{alpha_2 l_2} + alpha_3 beta_3 ⅇ^{alpha_3 l_3}) w 1}{lambda N p_prime_v 1 + k_P_vis} otherwise \end{matrix} \frac{d}{d time} (v) = phi_chi

Component: PE_velocity

k_S_vis = \{\begin{matrix} beta_S_lengthening ⅇ^{alpha_S_lengthening (l_2 - l_1)} if dl_2_dt ≦ dl_1_dt \\ beta_S_shortening ⅇ^{alpha_S_shortening (l_2 - l_1)} otherwise \end{matrix} \frac{d}{d time} (w) = \{\begin{matrix} \frac{k_S_vis (phi_chi - (alpha_S {(w - v)}^{2})) - (alpha_1 beta_1 ⅇ^{alpha_1 (l_2 - l_1)} (w - v) 1) - (alpha_2 beta_2 ⅇ^{alpha_2 l_2} w 1)}{k_S_vis} if isotonic_mode = 1 \\ \frac{k_S_vis (phi_chi - (alpha_S {(w - v)}^{2})) - (alpha_1 beta_1 ⅇ^{alpha_1 (l_2 - l_1)} (w - v) 1) - ((alpha_2 beta_2 ⅇ^{alpha_2 l_2} + alpha_3 beta_3 ⅇ^{alpha_3 l_3}) w 1)}{k_S_vis} otherwise \end{matrix}

Component: average_crossbridge_force

gamma = \frac{a d_h {(\frac{v_1}{v_max})}^{2}}{3 a d_h - (\frac{(a + 1) v}{v_max})} P_star = \{\begin{matrix} \frac{a (1 + \frac{v}{v_max})}{a - (\frac{v}{v_max})} if v ≦ 0 \\ 1 + d_h - (\frac{{d_h}^{2} a}{\frac{a d_h}{gamma} {(\frac{v}{v_max})}^{2} + \frac{(a + 1) v}{v_max} + a d_h}) otherwise \end{matrix} G_star = \{\begin{matrix} 1 + \frac{0.6 v}{v_max} if - v_max ≦ v \land v ≦ 0 \\ \frac{P_star}{\frac{\frac{0.4 a + 1}{a} v}{v_max} + 1} if 0 < v \land v ≦ v_1 \\ \frac{P_star ⅇ^{(- alpha_G) {(\frac{v - v_1}{v_max})}^{alpha_PG}}}{\frac{\frac{0.4 a + 1}{a} v}{v_max} + 1} otherwise \end{matrix} case_1 = \frac{a (0.4 + 0.4 a)}{v_max {((a + 1) 0.4)}^{2}} case_2 = \frac{a 1 (1 + 0.4 a + \frac{1.2 v}{v_max} + 0.6 {(\frac{v}{v_max})}^{2})}{v_max {((a - (\frac{v}{v_max})) (1 + \frac{0.6 v}{v_max}))}^{2}} case_3 = \frac{0.4 a + 1}{a v_max} case_4 = \frac{1}{v_max} ⅇ^{(- alpha_G) {(\frac{v}{v_max} - (\frac{v_1}{v_max}))}^{alpha_PG}} (\frac{0.4 a + 1}{a} + alpha_G alpha_P (1 + \frac{(0.4 a + 1) v}{a v_max})) {(\frac{v}{v_max} - (\frac{v_1}{v_max}))}^{alpha_PG - 1} p_v = \frac{P_star}{G_star} p_prime_v = \{\begin{matrix} case_1 phi_chi if v ≦ - v_max \\ case_2 phi_chi if - v_max < v \land v ≦ 0 \\ case_3 phi_chi if 0 < v \land v ≦ v_1 \\ case_4 phi_chi otherwise \end{matrix}

Component: calcium_handling

\frac{d}{d time} (A) = dA_dt dA_dt = a_on (A_tot - A) Ca_C - (a_off ⅇ^{(- k_A) A} pi_N_A A) N_A = \frac{N}{L_oz A} pi_N_A = \{\begin{matrix} 1 if N_A \geq 1 \\ {0.02}^{N_A} otherwise \end{matrix} \frac{d}{d time} (B) = dB_dt dB_dt = b_on (B_tot - B) Ca_C - (b_off B) \frac{d}{d time} (Ca_C) = \{\begin{matrix} b_c time (1 - (ⅇ^{(- a_c) {time}^{2}})) ⅇ^{(- a_c) {time}^{2}} if time < t_d \\ - dA_dt - dB_dt - (r_Ca ⅇ^{(- q_Ca) Ca_C} Ca_C) otherwise \end{matrix}

Source: Derived from workspace Katsnelson, Nikitina, Chemla, Solovyova, Coirault, Lecarpentier, Markhasin, 2004 at changeset 5725a8a06502.; This exposure was expired. A more up-to-date exposure is available, or view related resources.

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Influence of viscosity on myocardium mechanical activity: a mathematical model