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Model Mathematics

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Component: environment

Component: intracellular_ion_concentrations

Ca_b = Ca_TRPN_Max - TRPN \frac{d}{d time} (TRPN) = J_TRPN Ca_i = \{\begin{matrix} 1000 1.8433 e -7 if time < 1 \\ 1000 ((1.055 {(\frac{time}{1000})}^{3} - (0.03507 {(\frac{time}{1000})}^{2})) + \frac{0.0003992 time}{1000} - 1.356 e -6) if time \geq 10 \land time < 15 \\ 1000 ((0.014 {(\frac{time}{1000})}^{3} - (0.002555 {(\frac{time}{1000})}^{2})) + \frac{0.0001494 time}{1000} - 1.428 e -6) if time \geq 15 \land time < 55 \\ 1000 ((1.739 e -5 {(\frac{time}{1000})}^{3} - (3.209 e -6 {(\frac{time}{1000})}^{2}) - (\frac{5.689 e -6 time}{1000})) + 1.719 e -6) if time \geq 55 \land time < 250 \\ 1000 (((0.0001321 {(\frac{time}{1000})}^{4} - (0.0002197 {(\frac{time}{1000})}^{3})) + 0.0001374 {(\frac{time}{1000})}^{2} - (\frac{3.895 e -5 time}{1000})) + 4.441 e -6) if time \geq 250 \land time < 490 \\ 1000 1.2148 e -7 otherwise \end{matrix}

Component: thinfilaments

Component: tropomyosin

K_2 = \frac{alpha_r2 {z_p}^{n_Rel}}{{z_p}^{n_Rel} + {K_z}^{n_Rel}} (1 - (\frac{n_Rel {K_z}^{n_Rel}}{{z_p}^{n_Rel} + {K_z}^{n_Rel}})) K_1 = \frac{alpha_r2 {z_p}^{n_Rel - 1} n_Rel {K_z}^{n_Rel}}{{({z_p}^{n_Rel} + {K_z}^{n_Rel})}^{2}} z_max = \frac{\frac{alpha_0}{{(\frac{Ca_TRPN_50}{Ca_TRPN_Max})}^{n_Hill}} - K_2}{alpha_r1 + K_1 + \frac{alpha_0}{{(\frac{Ca_TRPN_50}{Ca_TRPN_Max})}^{n_Hill}}} Ca_50 = Ca_50ref (1 + beta_1 (lambda - 1)) Ca_TRPN_50 = \frac{Ca_50 Ca_TRPN_Max}{Ca_50 + \frac{k_Ref_off}{k_on} (1 - (\frac{(1 + beta_0 (lambda - 1)) 0.5}{gamma_trpn}))} alpha_Tm = alpha_0 {(\frac{Ca_b}{Ca_TRPN_50})}^{n_Hill} beta_Tm = alpha_r1 + \frac{alpha_r2 z^{n_Rel - 1}}{z^{n_Rel} + {K_z}^{n_Rel}} \frac{d}{d time} (z) = alpha_Tm (1 - z) - (beta_Tm z)

Component: troponin

k_off = \{\begin{matrix} k_Ref_off (1 - (\frac{Tension}{gamma_trpn T_ref})) if 1 - (\frac{Tension}{gamma_trpn T_ref}) > 0.1 \\ k_Ref_off 0.1 otherwise \end{matrix} J_TRPN = (Ca_TRPN_Max - TRPN) k_off - (Ca_i TRPN k_on)

Component: Myofilaments

ExtensionRatio = \{\begin{matrix} 1 if time > 1000 \\ 1 otherwise \end{matrix} lambda_prev = ExtensionRatio dExtensionRatiodt = 0 lambda = \{\begin{matrix} ExtensionRatio if ExtensionRatio > 0.8 \land ExtensionRatio ≦ 1.15 \\ 1.15 if ExtensionRatio > 1.15 \\ 0.8 otherwise \end{matrix}

Component: filament_overlap

overlap = 1 + beta_0 (lambda - 1)

Component: length_independent_tension

T_Base = \frac{T_ref z}{z_max}

Component: isometric_tension

T_0 = T_Base overlap

Component: Cross_Bridges

Q = Q_1 + Q_2 + Q_3 Tension = \{\begin{matrix} \frac{T_0 (a Q + 1)}{1 - Q} if Q < 0 \\ \frac{T_0 (1 + (a + 2) Q)}{1 + Q} otherwise \end{matrix} \frac{d}{d time} (Q_1) = A_1 dExtensionRatiodt - (alpha_1 Q_1) \frac{d}{d time} (Q_2) = A_2 dExtensionRatiodt - (alpha_2 Q_2) \frac{d}{d time} (Q_3) = A_3 dExtensionRatiodt - (alpha_3 Q_3)

Source: Derived from workspace Niederer, Hunter, Smith, 2006 at changeset 025af60a3d05.

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A quantitative analysis of cardiac myocyte relaxation: a simulation study