Model Mathematics

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Component: gas_exchange

x_m = \frac{(\log_{10} Pin_m - k1_m) + k2_m (pH_m - 7.4)}{k3_m} x_m1 = 10^{x_m} S_m = \frac{100 x_m1}{1 + x_m1} x_f = \frac{(\log_{10} Pin_f - k1_f) + k2_f (pH_f - 7.4)}{k3_f} x_f1 = 10^{x_f} S_f = \frac{100 x_f1}{1 + x_f1} O2_m = \frac{\frac{1.34 Hb_m}{100}}{k4_m {Pin_m}^{\frac{- 1}{k3_m}} + 1} + a Pin_m O2_f = \frac{\frac{1.34 Hb_f}{100}}{k4_f {Pin_f}^{\frac{- 1}{k3_f}} + 1} + a Pin_f x1_m = \{\begin{matrix} ⅇ^{(- 1) c2 (100 - 96)} if S_m > 96 \\ ⅇ^{(- 1) c2 (100 - S_m)} otherwise \end{matrix} theta_96_m = \frac{\frac{\frac{(c1 (1 - x1_m) + c3 S_m - c4) 1.34}{100} Hb_m}{0.2} ks_m}{kh} theta_m = \{\begin{matrix} theta_96_m - (\frac{theta_96_m (96 - S_m)}{- 4}) if S_m > 96 \\ \frac{\frac{\frac{(c1 (1 - x1_m) + c3 S_m - c4) 1.34}{100} Hb_m}{0.2} ks_m}{kh} otherwise \end{matrix} x1_f = \{\begin{matrix} ⅇ^{(- 1) c2 (100 - 96)} if S_f > 96 \\ ⅇ^{(- 1) c2 (100 - S_f)} otherwise \end{matrix} theta_96_f = \frac{\frac{\frac{(c1 (1 - x1_f) + c3 S_f - c4) 1.34}{100} Hb_f}{0.2} ks_f}{kh} theta_f = \{\begin{matrix} theta_96_f - (\frac{theta_96_f (96 - S_f)}{- 4}) if S_f > 96 \\ \frac{\frac{\frac{(c1 (1 - x1_f) + c3 S_f - c4) 1.34}{100} Hb_f}{0.2} ks_f}{kh} otherwise \end{matrix} R_m = \frac{1}{theta_m V_m} R_f = \frac{1}{theta_f V_f} R_p = \frac{1}{Dm} R_t = R_m + R_f + R_p Dp = \frac{\frac{1}{R_t}}{60} dodt_m = \frac{B (- 1) Dp (Pin_m - Pin_f)}{V_m} dodt_f = \frac{\frac{- 1}{B} V_m}{V_f} dodt_m tt = \frac{V_f}{\frac{Q_f}{60}} dt = \frac{tt}{100} O2_mnew = O2_m + \frac{dodt_m dt}{2} O2_fnew = O2_f + \frac{dodt_f dt}{2} f_m = \frac{\frac{1.34 Hb_m}{100}}{k4_m {Pin_m}^{\frac{- 1}{k3_m}} + 1} + a Pin_m - O2_mnew f_f = \frac{\frac{1.34 Hb_f}{100}}{k4_f {Pin_f}^{\frac{- 1}{k3_f}} + 1} + a Pin_f - O2_fnew upper_m = 1.34 Hb_m k4_m {Pin_m}^{\frac{- 1}{k3_m} - 1} lower_m = 100 k3_m {(k4_m {Pin_m}^{\frac{- 1}{k3_m}} + 1)}^{2} dfdP_m = \frac{upper_m}{lower_m} + a upper_f = 1.34 Hb_f k4_f {Pin_f}^{\frac{- 1}{k3_f} - 1} lower_f = 100 k3_f {(k4_f {Pin_f}^{\frac{- 1}{k3_f}} + 1)}^{2} dfdP_f = \frac{upper_f}{lower_f} + a Pend1_m = Pin_m - (\frac{f_m}{dfdP_m}) Pend1_f = Pin_f - (\frac{f_f}{dfdP_f}) x_m_sec = \frac{(\log_{10} Pend1_m - k1_m) + k2_m (pH_m - 7.4)}{k3_m} x_m1_sec = 10^{x_m_sec} S_m_sec = \frac{100 x_m1_sec}{1 + x_m1_sec} x_f_sec = \frac{(\log_{10} Pend1_f - k1_f) + k2_f (pH_f - 7.4)}{k3_f} x_f1_sec = 10^{x_f_sec} S_f_sec = \frac{100 x_f1_sec}{1 + x_f1_sec} x1_m_sec = \{\begin{matrix} ⅇ^{(- 1) c2 (100 - 96)} if S_m_sec > 96 \\ ⅇ^{(- 1) c2 (100 - S_m_sec)} otherwise \end{matrix} theta_96_m_sec = \frac{\frac{\frac{(c1 (1 - x1_m_sec) + c3 S_m_sec - c4) 1.34}{100} Hb_m}{0.2} ks_m}{kh} theta_m_sec = \{\begin{matrix} theta_96_m_sec - (\frac{theta_96_m_sec (96 - S_m_sec)}{- 4}) if S_m_sec > 96 \\ \frac{\frac{\frac{(c1 (1 - x1_m_sec) + c3 S_m_sec - c4) 1.34}{100} Hb_m}{0.2} ks_m}{kh} otherwise \end{matrix} x1_f_sec = \{\begin{matrix} ⅇ^{(- 1) c2 (100 - 96)} if S_f_sec > 96 \\ ⅇ^{(- 1) c2 (100 - S_f_sec)} otherwise \end{matrix} theta_96_f_sec = \frac{\frac{\frac{(c1 (1 - x1_f_sec) + c3 S_f_sec - c4) 1.34}{100} Hb_f}{0.2} ks_f}{kh} theta_f_sec = \{\begin{matrix} theta_96_f_sec - (\frac{theta_96_f_sec (96 - S_f_sec)}{- 4}) if S_f_sec > 96 \\ \frac{\frac{\frac{(c1 (1 - x1_f_sec) + c3 S_f_sec - c4) 1.34}{100} Hb_f}{0.2} ks_f}{kh} otherwise \end{matrix} R_m_sec = \frac{1}{theta_m_sec V_m} R_f_sec = \frac{1}{theta_f_sec V_f} R_p_sec = \frac{1}{Dm} R_t_sec = R_m_sec + R_f_sec + R_p_sec Dp_sec = \frac{\frac{1}{R_t_sec}}{60} dodt_m_sec = \frac{B (- 1) Dp_sec (Pend1_m - Pend1_f)}{V_m} dodt_f_sec = \frac{\frac{- 1}{B} V_m}{V_f} dodt_m_sec O2_mnew_sec = O2_m + \frac{dodt_m_sec dt}{2} O2_fnew_sec = O2_f + \frac{dodt_f_sec dt}{2} f_m_sec = \frac{\frac{1.34 Hb_m}{100}}{k4_m {Pend1_m}^{\frac{- 1}{k3_m}} + 1} + a Pend1_m - O2_mnew_sec f_f_sec = \frac{\frac{1.34 Hb_f}{100}}{k4_f {Pend1_f}^{\frac{- 1}{k3_f}} + 1} + a Pend1_f - O2_fnew_sec upper_m_sec = 1.34 Hb_m k4_m {Pend1_m}^{\frac{- 1}{k3_m} - 1} lower_m_sec = 100 k3_m {(k4_m {Pend1_m}^{\frac{- 1}{k3_m}} + 1)}^{2} dfdP_m_sec = \frac{upper_m_sec}{lower_m_sec} + a upper_f_sec = 1.34 Hb_f k4_f {Pend1_f}^{\frac{- 1}{k3_f} - 1} lower_f_sec = 100 k3_f {(k4_f {Pend1_f}^{\frac{- 1}{k3_f}} + 1)}^{2} dfdP_f_sec = \frac{upper_f_sec}{lower_f_sec} + a Pend2_m = Pend1_m - (\frac{f_m_sec}{dfdP_m_sec}) Pend2_f = Pend1_f - (\frac{f_f_sec}{dfdP_f_sec}) x_m_trd = \frac{(\log_{10} Pend2_m - k1_m) + k2_m (pH_m - 7.4)}{k3_m} x_m1_trd = 10^{x_m_trd} S_m_trd = \frac{100 x_m1_trd}{1 + x_m1_trd} x_f_trd = \frac{(\log_{10} Pend2_f - k1_f) + k2_f (pH_f - 7.4)}{k3_f} x_f1_trd = 10^{x_f_trd} S_f_trd = \frac{100 x_f1_trd}{1 + x_f1_trd} x1_m_trd = \{\begin{matrix} ⅇ^{(- 1) c2 (100 - 96)} if S_m_trd > 96 \\ ⅇ^{(- 1) c2 (100 - S_m_trd)} otherwise \end{matrix} theta_96_m_trd = \frac{\frac{\frac{(c1 (1 - x1_m_trd) + c3 S_m_trd - c4) 1.34}{100} Hb_m}{0.2} ks_m}{kh} theta_m_trd = \{\begin{matrix} theta_96_m_trd - (\frac{theta_96_m_trd (96 - S_m_trd)}{- 4}) if S_m_trd > 96 \\ \frac{\frac{\frac{(c1 (1 - x1_m_trd) + c3 S_m_trd - c4) 1.34}{100} Hb_m}{0.2} ks_m}{kh} otherwise \end{matrix} x1_f_trd = \{\begin{matrix} ⅇ^{(- 1) c2 (100 - 96)} if S_f_trd > 96 \\ ⅇ^{(- 1) c2 (100 - S_f_trd)} otherwise \end{matrix} theta_96_f_trd = \frac{\frac{\frac{(c1 (1 - x1_f_trd) + c3 S_f_trd - c4) 1.34}{100} Hb_f}{0.2} ks_f}{kh} theta_f_trd = \{\begin{matrix} theta_96_f_trd - (\frac{theta_96_f_trd (96 - S_f_trd)}{- 4}) if S_f_trd > 96 \\ \frac{\frac{\frac{(c1 (1 - x1_f_trd) + c3 S_f_trd - c4) 1.34}{100} Hb_f}{0.2} ks_f}{kh} otherwise \end{matrix} R_m_trd = \frac{1}{theta_m_trd V_m} R_f_trd = \frac{1}{theta_f_trd V_f} R_p_trd = \frac{1}{Dm} R_t_trd = R_m_trd + R_f_trd + R_p_trd Dp_trd = \frac{\frac{1}{R_t_trd}}{60} dodt_m_trd = \frac{B (- 1) Dp_trd (Pend2_m - Pend2_f)}{V_m} dodt_f_trd = \frac{\frac{- 1}{B} V_m}{V_f} dodt_m_trd O2_mnew_trd = O2_m + dodt_m_trd dt O2_fnew_trd = O2_f + dodt_f_trd dt f_m_trd = \frac{\frac{1.34 Hb_m}{100}}{k4_m {Pend2_m}^{\frac{- 1}{k3_m}} + 1} + a Pend2_m - O2_mnew_trd f_f_trd = \frac{\frac{1.34 Hb_f}{100}}{k4_f {Pend2_f}^{\frac{- 1}{k3_f}} + 1} + a Pend2_f - O2_fnew_trd upper_m_trd = 1.34 Hb_m k4_m {Pend2_m}^{\frac{- 1}{k3_m} - 1} lower_m_trd = 100 k3_m {(k4_m {Pend2_m}^{\frac{- 1}{k3_m}} + 1)}^{2} dfdP_m_trd = \frac{upper_m_trd}{lower_m_trd} + a upper_f_trd = 1.34 Hb_f k4_f {Pend2_f}^{\frac{- 1}{k3_f} - 1} lower_f_trd = 100 k3_f {(k4_f {Pend2_f}^{\frac{- 1}{k3_f}} + 1)}^{2} dfdP_f_trd = \frac{upper_f_trd}{lower_f_trd} + a Pend3_m = Pend2_m - (\frac{f_m_trd}{dfdP_m_trd}) Pend3_f = Pend2_f - (\frac{f_f_trd}{dfdP_f_trd}) x_m_for = \frac{(\log_{10} Pend3_m - k1_m) + k2_m (pH_m - 7.4)}{k3_m} x_m1_for = 10^{x_m_for} S_m_for = \frac{100 x_m1_for}{1 + x_m1_for} x_f_for = \frac{(\log_{10} Pend3_f - k1_f) + k2_f (pH_f - 7.4)}{k3_f} x_f1_for = 10^{x_f_for} S_f_for = \frac{100 x_f1_for}{1 + x_f1_for} x1_m_for = \{\begin{matrix} ⅇ^{(- 1) c2 (100 - 96)} if S_m_for > 96 \\ ⅇ^{(- 1) c2 (100 - S_m_for)} otherwise \end{matrix} theta_96_m_for = \frac{\frac{\frac{(c1 (1 - x1_m_for) + c3 S_m_for - c4) 1.34}{100} Hb_m}{0.2} ks_m}{kh} theta_m_for = \{\begin{matrix} theta_96_m_for - (\frac{theta_96_m_for (96 - S_m_for)}{- 4}) if S_m_for > 96 \\ \frac{\frac{\frac{(c1 (1 - x1_m_for) + c3 S_m_for - c4) 1.34}{100} Hb_m}{0.2} ks_m}{kh} otherwise \end{matrix} x1_f_for = \{\begin{matrix} ⅇ^{(- 1) c2 (100 - 96)} if S_f_for > 96 \\ ⅇ^{(- 1) c2 (100 - S_f_for)} otherwise \end{matrix} theta_96_f_for = \frac{\frac{\frac{(c1 (1 - x1_f_for) + c3 S_f_for - c4) 1.34}{100} Hb_f}{0.2} ks_f}{kh} theta_f_for = \{\begin{matrix} theta_96_f_for - (\frac{theta_96_f_for (96 - S_f_for)}{- 4}) if S_f_for > 96 \\ \frac{\frac{\frac{(c1 (1 - x1_f_for) + c3 S_f_for - c4) 1.34}{100} Hb_f}{0.2} ks_f}{kh} otherwise \end{matrix} R_m_for = \frac{1}{theta_m_for V_m} R_f_for = \frac{1}{theta_f_for V_f} R_p_for = \frac{1}{Dm} R_t_for = R_m_for + R_f_for + R_p_for Dp_for = \frac{\frac{1}{R_t_for}}{60} dodt_m_for = \frac{B (- 1) Dp_for (Pend3_m - Pend3_f)}{V_m} dodt_f_for = \frac{\frac{- 1}{B} V_m}{V_f} dodt_m_for O2_mnew_final = O2_m + \frac{1}{6} (dodt_m + 2 dodt_m_sec + 2 dodt_m_trd + dodt_m_for) dt O2_fnew_final = O2_f + \frac{1}{6} (dodt_f + 2 dodt_f_sec + 2 dodt_f_trd + dodt_f_for) dt f_m_final = \frac{\frac{1.34 Hb_m}{100}}{k4_m {Pend3_m}^{\frac{- 1}{k3_m}} + 1} + a Pend3_m - O2_mnew_final f_f_final = \frac{\frac{1.34 Hb_f}{100}}{k4_f {Pend3_f}^{\frac{- 1}{k3_f}} + 1} + a Pend3_f - O2_fnew_final upper_m_final = 1.34 Hb_m k4_m {Pend3_m}^{\frac{- 1}{k3_m} - 1} lower_m_final = 100 k3_m {(k4_m {Pend3_m}^{\frac{- 1}{k3_m}} + 1)}^{2} dfdP_m_final = \frac{upper_m_final}{lower_m_final} + a upper_f_final = 1.34 Hb_f k4_f {Pend3_f}^{\frac{- 1}{k3_f} - 1} lower_f_final = 100 k3_f {(k4_f {Pend3_f}^{\frac{- 1}{k3_f}} + 1)}^{2} dfdP_f_final = \frac{upper_f_final}{lower_f_final} + a Pfinal_m = Pend3_m - (\frac{f_m_final}{dfdP_m_final}) Pfinal_f = Pend3_f - (\frac{f_f_final}{dfdP_f_final}) \frac{d}{d t} (Pin_m) = \frac{Pfinal_m - Pin_m}{dt} \frac{d}{d t} (Pin_f) = \frac{Pfinal_f - Pin_f}{dt}