Model Mathematics

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theta_rad = \frac{theta_deg π}{180}

z = \{\begin{matrix} 1 if t > 100 \land t < 101 \\ 0 otherwise \end{matrix}

mt = \frac{t}{T} - (⌊ \frac{t}{T} ⌋) mta = \frac{t}{T} - delay - (⌊ \frac{t - (delay T)}{T} ⌋)

E_B_LV = \{\begin{matrix} \frac{(E_B_LVmax - E_B_LVmin) mt}{To_pk} ⅇ^{0.5 (1 - ({(\frac{mt}{To_pk})}^{2}))} + E_B_LVmin if mt < 0.4 \\ \frac{(E_B_LVmax - E_B_LVmin) 0.4}{To_pk} ⅇ^{0.5 (1 - ({(\frac{0.4}{To_pk})}^{2}))} + E_B_LVmin if mt < 0.6 \\ E_B_LVmin otherwise \end{matrix} \frac{d}{d t} (q_B_LV) = v_B_MV - v_B_AV u_B_LV = E_B_LV (q_B_LV - q_B_LV_us)

E_B_RV = \{\begin{matrix} \frac{(E_B_RVmax - E_B_RVmin) mt}{To_pk} ⅇ^{0.5 (1 - ({(\frac{mt}{To_pk})}^{2}))} + E_B_RVmin if mt < 0.4 \\ \frac{(E_B_RVmax - E_B_RVmin) 0.4}{To_pk} ⅇ^{0.5 (1 - ({(\frac{0.4}{To_pk})}^{2}))} + E_B_RVmin if mt < 0.6 \\ E_B_RVmin otherwise \end{matrix} \frac{d}{d t} (q_B_RV) = v_B_TV - v_B_PV u_B_RV = E_B_RV (q_B_RV - q_B_RV_us)

\frac{d}{d t} (q_B_LA) = v_B_pulmVein - v_B_MV u_B_LA = E_B_LA (q_B_LA - q_B_LA_us)

\frac{d}{d t} (q_B_RA) = v_B_brain3 + v_B_liver2 + v_B_lRenal3 + v_B_rRenal3 + v_B_musTorso3 + v_B_lArm3 + v_B_rArm3 + v_B_lLeg3 + v_B_rLeg3 - v_B_TV u_B_RA = E_B_RA (q_B_RA - q_B_RA_us)

v_B_AV = \{\begin{matrix} \frac{u_B_LV - u_B_AA}{R_B_AV} if u_B_LV > u_B_AA \\ 0 otherwise \end{matrix}

v_B_PV = \{\begin{matrix} \frac{u_B_RV - u_B_PA}{R_B_PV} if u_B_RV > u_B_PA \\ 0 otherwise \end{matrix}

v_B_TV = \{\begin{matrix} \frac{u_B_RA - u_B_RV}{R_B_TV} if u_B_RA > u_B_RV \\ 0 otherwise \end{matrix}

v_B_MV = \{\begin{matrix} \frac{u_B_LA - u_B_LV}{R_B_MV} if u_B_LA > u_B_LV \\ 0 otherwise \end{matrix}

\frac{d}{d t} (q_B_PA) = v_B_PV - v_B_lung1 u_B_PA = E_B_PA (q_B_PA - q_B_PA_us)

\frac{d}{d t} (q_B_lung) = v_B_lung1 - v_B_lung2 u_B_lung = E_B_lung (q_B_lung - q_B_lung_us) - (rho g z_lung \cos theta_rad)

v_B_lung1 = \frac{u_B_PA - u_B_lung}{R_B_lung1} v_B_lung2 = \frac{u_B_lung - u_B_pulmVein}{R_B_lung2}

\frac{d}{d t} (q_B_pulmVein) = v_B_lung2 - v_B_pulmVein u_B_pulmVein = E_B_pulmVein (q_B_pulmVein - q_B_pulmVein_us)

v_B_pulmVein = \frac{u_B_pulmVein - u_B_LA}{R_B_pulmVein}

\frac{d}{d t} (q_B_brain) = v_B_brain1 - v_B_brain2 u_B_brain = E_B_brain (q_B_brain - q_B_brain_us) \frac{d}{d t} (q_B_brainVein) = v_B_brain2 - v_B_brain3 u_B_brainVein = E_B_brainVein (q_B_brainVein - q_B_brainVein_us)

v_B_brain1 = \frac{u_B_AA - u_B_brain}{R_B_brain1} v_B_brain2 = \frac{u_B_brain - u_B_brainVein}{R_B_brain2} v_B_brain3 = \frac{u_B_brainVein - u_B_RA}{R_B_brain3}

\frac{d}{d t} (q_B_AA) = v_B_AV - v_B_brain1 - v_B_celiac - v_B_supMes - v_B_infMes - v_B_lRenal1 - v_B_rRenal1 - v_B_musTorso1 - v_B_lArm1 - v_B_rArm1 - v_B_lLeg1 - v_B_rLeg1 u_B_AA = E_B_AA (q_B_AA - q_B_AA_us)

\frac{d}{d t} (q_B_celiac) = v_B_celiac - v_B_liver1 - v_B_stomach1 - v_B_spleen1 - v_B_pancreas1 u_B_celiac = E_B_celiac (q_B_celiac - q_B_celiac_us) - (rho g z_celiac \cos theta_rad)

v_B_celiac = \frac{u_B_AA - u_B_celiac}{R_B_celiac}

\frac{d}{d t} (q_B_supMes) = v_B_supMes - v_B_pancreas2 - v_B_intestine1 - v_B_colon1 u_B_supMes = E_B_supMes (q_B_supMes - q_B_supMes_us) - (rho g z_supMes \cos theta_rad)

v_B_supMes = \frac{u_B_AA - u_B_supMes}{R_B_supMes}

v_B_infMes = \frac{u_B_AA - u_B_colon}{R_B_infMes}

\frac{d}{d t} (q_B_stomach) = v_B_stomach1 - v_B_stomach2 u_B_stomach = E_B_stomach (q_B_stomach - q_B_stomach_us) - (rho g z_stomach \cos theta_rad)

v_B_stomach1 = \frac{u_B_celiac - u_B_stomach}{R_B_stomach1} v_B_stomach2 = \frac{u_B_stomach - u_B_portalVein}{R_B_stomach2}

\frac{d}{d t} (q_B_spleen) = v_B_spleen1 - v_B_spleen2 u_B_spleen = E_B_colon (q_B_spleen - q_B_spleen_us) - (rho g z_spleen \cos theta_rad)

v_B_spleen1 = \frac{u_B_celiac - u_B_spleen}{R_B_spleen1} v_B_spleen2 = \frac{u_B_spleen - u_B_portalVein}{R_B_spleen2}

\frac{d}{d t} (q_B_pancreas) = v_B_pancreas1 + v_B_pancreas2 - v_B_pancreas3 u_B_pancreas = E_B_pancreas (q_B_pancreas - q_B_pancreas_us) - (rho g z_pancreas \cos theta_rad)

v_B_pancreas1 = \frac{u_B_celiac - u_B_pancreas}{R_B_pancreas1} v_B_pancreas2 = \frac{u_B_supMes - u_B_pancreas}{R_B_pancreas2} v_B_pancreas3 = \frac{u_B_pancreas - u_B_portalVein}{R_B_pancreas3}

\frac{d}{d t} (q_B_intestine) = v_B_intestine1 - v_B_intestine2 u_B_intestine = E_B_intestine (q_B_intestine - q_B_intestine_us)

v_B_intestine1 = \frac{u_B_supMes - u_B_intestine}{R_B_intestine1} v_B_intestine2 = \frac{u_B_intestine - u_B_portalVein}{R_B_intestine2}

\frac{d}{d t} (q_B_colon) = v_B_colon1 + v_B_infMes - v_B_colon2 u_B_colon = E_B_colon (q_B_colon - q_B_colon_us) - (rho g z_colon \cos theta_rad)

v_B_colon1 = \frac{u_B_supMes - u_B_colon}{R_B_colon1} v_B_colon2 = \frac{u_B_colon - u_B_portalVein}{R_B_colon2}

\frac{d}{d t} (q_B_portalVein) = v_B_stomach2 + v_B_spleen2 + v_B_pancreas3 + v_B_intestine2 + v_B_colon2 - v_B_portalVein u_B_portalVein = E_B_portalVein (q_B_portalVein - q_B_portalVein_us) - (rho g z_portalVein \cos theta_rad)

v_B_portalVein = \frac{u_B_portalVein - u_B_liver}{R_B_portalVein}

\frac{d}{d t} (q_B_liver) = v_B_liver1 + v_B_portalVein - v_B_liver2 u_B_liver = E_B_liver (q_B_liver - q_B_liver_us) - (rho g z_liver \cos theta_rad)

v_B_liver1 = \frac{u_B_celiac - u_B_liver}{R_B_liver1} v_B_liver2 = \frac{u_B_liver - u_B_RA}{R_B_liver2}

\frac{d}{d t} (q_B_lGlomerulus) = v_B_lRenal1 - v_B_lRenal2 \frac{d}{d t} (q_B_rGlomerulus) = v_B_lRenal1 - v_B_lRenal2 \frac{d}{d t} (q_B_lPeritubular) = v_B_lRenal2 - v_B_lRenal3 \frac{d}{d t} (q_B_rPeritubular) = v_B_lRenal2 - v_B_lRenal3 u_B_lGlomerulus = E_B_lGlomerulus (q_B_lGlomerulus - q_B_lGlomerulus_us) - (rho g z_kidney \cos theta_rad) u_B_rGlomerulus = E_B_rGlomerulus (q_B_rGlomerulus - q_B_rGlomerulus_us) - (rho g z_kidney \cos theta_rad) u_B_lPeritubular = E_B_lPeritubular (q_B_lPeritubular - q_B_lPeritubular_us) - (rho g z_kidney \cos theta_rad) u_B_rPeritubular = E_B_rPeritubular (q_B_rPeritubular - q_B_rPeritubular_us) - (rho g z_kidney \cos theta_rad)

v_B_lRenal1 = \frac{u_B_AA - u_B_lGlomerulus}{R_B_lRenal1} v_B_rRenal1 = \frac{u_B_AA - u_B_rGlomerulus}{R_B_rRenal1} v_B_lRenal2 = \frac{u_B_lGlomerulus - u_B_lPeritubular}{R_B_lRenal2} v_B_rRenal2 = \frac{u_B_rGlomerulus - u_B_rPeritubular}{R_B_rRenal2} v_B_lRenal3 = \frac{u_B_lPeritubular - u_B_RA}{R_B_lRenal3} v_B_rRenal3 = \frac{u_B_rPeritubular - u_B_RA}{R_B_rRenal3}

\frac{d}{d t} (q_B_musTorso) = v_B_musTorso1 - v_B_musTorso2 u_B_musTorso = E_B_musTorso (q_B_musTorso - q_B_musTorso_us) - (rho g z_musTorso \cos theta_rad) \frac{d}{d t} (q_B_musTorsoVein) = v_B_musTorso2 - v_B_musTorso3 u_B_musTorsoVein = E_B_musTorsoVein (q_B_musTorsoVein - q_B_musTorsoVein_us) - (rho g z_musTorso \cos theta_rad)

v_B_musTorso1 = \frac{u_B_AA - u_B_musTorso}{R_B_musTorso1} v_B_musTorso2 = \frac{u_B_musTorso - u_B_musTorsoVein}{R_B_musTorso2} v_B_musTorso3 = \frac{u_B_musTorsoVein - u_B_RA}{R_B_musTorso3}

\frac{d}{d t} (q_B_lArm) = v_B_lArm1 - v_B_lArm2 u_B_lArm = E_B_lArm (q_B_lArm - q_B_lArm_us) - (rho g z_lArm \cos theta_rad) \frac{d}{d t} (q_B_lArmVein) = v_B_lArm2 - v_B_lArm3 u_B_lArmVein = E_B_lArmVein (q_B_lArmVein - q_B_lArmVein_us) - (rho g z_lArm \cos theta_rad)

v_B_lArm1 = \frac{u_B_AA - u_B_lArm}{R_B_lArm1} v_B_lArm2 = \frac{u_B_lArm - u_B_lArmVein}{R_B_lArm2} v_B_lArm3 = \frac{u_B_lArmVein - u_B_RA}{R_B_lArm3}

\frac{d}{d t} (q_B_rArm) = v_B_rArm1 - v_B_rArm2 u_B_rArm = E_B_rArm (q_B_rArm - q_B_rArm_us) - (rho g z_rArm \cos theta_rad) \frac{d}{d t} (q_B_rArmVein) = v_B_rArm2 - v_B_rArm3 u_B_rArmVein = E_B_rArmVein (q_B_rArmVein - q_B_rArm_us) - (rho g z_rArm \cos theta_rad)

v_B_rArm1 = \frac{u_B_AA - u_B_rArm}{R_B_rArm1} v_B_rArm2 = \frac{u_B_rArm - u_B_rArmVein}{R_B_rArm2} v_B_rArm3 = \frac{u_B_rArmVein - u_B_RA}{R_B_rArm3}

\frac{d}{d t} (q_B_lLeg) = v_B_lLeg1 - v_B_lLeg2 u_B_lLeg = E_B_lLeg (q_B_lLeg - q_B_lLeg_us) - (rho g z_lLeg \cos theta_rad) \frac{d}{d t} (q_B_lLegVein) = v_B_lLeg2 - v_B_lLeg3 u_B_lLegVein = E_B_lLegVein (q_B_lLegVein - q_B_lLeg_us) - (rho g z_lLeg \cos theta_rad)

v_B_lLeg1 = \frac{u_B_AA - u_B_lLeg}{R_B_lLeg1} v_B_lLeg2 = \frac{u_B_lLeg - u_B_lLegVein}{R_B_lLeg2} v_B_lLeg3 = \frac{u_B_lLegVein - u_B_RA}{R_B_lLeg3}

\frac{d}{d t} (q_B_rLeg) = v_B_rLeg1 - v_B_rLeg2 u_B_rLeg = E_B_rLeg (q_B_rLeg - q_B_rLeg_us) - (rho g z_rLeg \cos theta_rad) \frac{d}{d t} (q_B_rLegVein) = v_B_rLeg2 - v_B_rLeg3 u_B_rLegVein = E_B_rLegVein (q_B_rLegVein - q_B_rLeg_us) - (rho g z_rLeg \cos theta_rad)

v_B_rLeg1 = \frac{u_B_AA - u_B_rLeg}{R_B_rLeg1} v_B_rLeg2 = \frac{u_B_rLeg - u_B_rLegVein}{R_B_rLeg2} v_B_rLeg3 = \frac{u_B_rLegVein - u_B_RA}{R_B_rLeg3}

q_B_heartTot = q_B_LV + q_B_RV + q_B_LA + q_B_RA + q_B_AA + q_B_PA q_B_lungTot = q_B_lung + q_B_pulmVein q_B_renalTot = q_B_lGlomerulus + q_B_rGlomerulus + q_B_lPeritubular + q_B_rPeritubular q_B_gutTot = q_B_celiac + q_B_supMes + q_B_stomach + q_B_spleen + q_B_pancreas + q_B_intestine + q_B_colon + q_B_portalVein + q_B_liver + q_B_musTorso + q_B_musTorsoVein q_B_brainTot = q_B_brain + q_B_brainVein q_B_limbTot = q_B_lArm + q_B_lArmVein + q_B_rArm + q_B_rArmVein + q_B_lLeg + q_B_lLegVein + q_B_rLeg + q_B_rLegVein q_Blood = q_B_heartTot + q_B_lungTot + q_B_renalTot + q_B_gutTot + q_B_brainTot + q_B_limbTot