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_LV = \{\begin{matrix} \frac{(E_LVmax - E_LVmin) mt}{To_pk} ⅇ^{0.5 (1 - ({(\frac{mt}{To_pk})}^{2}))} + E_LVmin if mt < 0.4 \\ \frac{(E_LVmax - E_LVmin) 0.4}{To_pk} ⅇ^{0.5 (1 - ({(\frac{0.4}{To_pk})}^{2}))} + E_LVmin if mt < 0.6 \\ E_LVmin otherwise \end{matrix} \frac{d}{d t} (q_LV) = v_MV - v_AV u_LV = E_LV (q_LV - q_LV_us)

E_RV = \{\begin{matrix} \frac{(E_RVmax - E_RVmin) mt}{To_pk} ⅇ^{0.5 (1 - ({(\frac{mt}{To_pk})}^{2}))} + E_RVmin if mt < 0.4 \\ \frac{(E_RVmax - E_RVmin) 0.4}{To_pk} ⅇ^{0.5 (1 - ({(\frac{0.4}{To_pk})}^{2}))} + E_RVmin if mt < 0.6 \\ E_RVmin otherwise \end{matrix} \frac{d}{d t} (q_RV) = v_TV - v_PV u_RV = E_RV (q_RV - q_RV_us)

\frac{d}{d t} (q_LA) = v_pulmV - v_MV u_LA = E_LA (q_LA - q_LA_us)

\frac{d}{d t} (q_RA) = v_headV + v_lArmV + v_rArmV + v_V1 - v_TV u_RA = E_RA (q_RA - q_RA_us)

v_AV = \{\begin{matrix} \frac{u_LV - u_AA}{R_AV} if u_LV > u_AA \\ 0 otherwise \end{matrix}

v_PV = \{\begin{matrix} \frac{u_RV - u_pulmA}{R_PV} if u_RV > u_pulmA \\ 0 otherwise \end{matrix}

v_TV = \{\begin{matrix} \frac{u_RA - u_RV}{R_TV} if u_RA > u_RV \\ 0 otherwise \end{matrix}

v_MV = \{\begin{matrix} \frac{u_LA - u_LV}{R_MV} if u_LA > u_LV \\ 0 otherwise \end{matrix}

\frac{d}{d t} (q_pulmA) = v_PV - v_lungA u_pulmA = E_pulmA (q_pulmA - q_pulmA_us)

v_lungA = \frac{u_pulmA - u_lungC}{R_lungA}

\frac{d}{d t} (q_lungC) = v_lungA - v_pulmV u_lungC = E_lungC (q_lungC - q_lungC_us)

v_pulmV = \frac{u_lungC - u_LA}{R_pulmV}

\frac{d}{d t} (q_AA) = v_AV - v_A1 - v_brachiocephalicA - v_lCCA - v_lSubclavianA u_AA = E_AA (q_AA - q_AA_us)

v_brachiocephalicA = \frac{u_AA - u_brachiocephalicA}{R_brachiocephalicA}

\frac{d}{d t} (q_brachiocephalicA) = v_brachiocephalicA - v_rSubclavianA - v_rCCA u_brachiocephalicA = E_brachiocephalicA (q_brachiocephalicA - q_brachiocephalicA_us)

v_rSubclavianA = \frac{u_brachiocephalicA - u_rSubclavianA}{R_rSubclavianA}

v_rCCA = \frac{u_brachiocephalicA - u_headC}{R_rCCA}

\frac{d}{d t} (q_rSubclavianA) = v_rSubclavianA - v_rBrachialA - v_rVertebralA u_rSubclavianA = E_rSubclavianA (q_rSubclavianA - q_rSubclavianA_us)

v_rBrachialA = \frac{u_rSubclavianA - u_rArmC}{R_rBrachialA}

v_rVertebralA = \frac{u_rSubclavianA - u_headC}{R_rVertebralA}

v_lCCA = \frac{u_AA - u_headC}{R_lCCA}

v_lSubclavianA = \frac{u_AA - u_lSubclavianA}{R_lSubclavianA}

\frac{d}{d t} (q_lSubclavianA) = v_lSubclavianA - v_lBrachialA - v_lVertebralA u_lSubclavianA = E_lSubclavianA (q_lSubclavianA - q_lSubclavianA_us)

v_lBrachialA = \frac{u_lSubclavianA - u_lArmC}{R_lBrachialA}

v_lVertebralA = \frac{u_lSubclavianA - u_headC}{R_lVertebralA}

\frac{d}{d t} (q_headC) = v_rVertebralA + v_rCCA + v_lCCA + v_lVertebralA - v_headV u_headC = E_headC (q_headC - q_headC_us)

v_headV = \frac{u_headC - u_RA}{R_headV}

\frac{d}{d t} (q_lArmC) = v_lBrachialA - v_lArmV u_lArmC = E_lArmC (q_lArmC - q_lArmC_us)

v_lArmV = \frac{u_lArmC - u_RA}{R_lArmV}

\frac{d}{d t} (q_rArmC) = v_rBrachialA - v_rArmV u_rArmC = E_rArmC (q_rArmC - q_rArmC_us)

v_rArmV = \frac{u_rArmC - u_RA}{R_rArmV}

\frac{d}{d t} (q_A1) = v_A1 - v_A2 - v_lInfPhrenicA \frac{d}{d t} (q_A2) = v_A2 - v_A3 - v_rInfPhrenicA \frac{d}{d t} (q_A3) = v_A3 - v_A4 - v_celiacA \frac{d}{d t} (q_A4) = v_A4 - v_A5 - v_supMesA \frac{d}{d t} (q_A5) = v_A5 - v_A6 - v_lRenalA \frac{d}{d t} (q_A6) = v_A6 - v_A7 - v_rRenalA \frac{d}{d t} (q_A7) = v_A7 - v_A8 - v_infMesA \frac{d}{d t} (q_A8) = v_A8 - v_lCommonIliacA - v_rCommonIliacA

u_A1 = E_A1 (q_A1 - q_A1_us) u_A2 = E_A2 (q_A2 - q_A2_us) u_A3 = E_A3 (q_A3 - q_A3_us) u_A4 = E_A4 (q_A4 - q_A4_us) u_A5 = E_A5 (q_A5 - q_A5_us) u_A6 = E_A6 (q_A6 - q_A6_us) u_A7 = E_A7 (q_A7 - q_A7_us) u_A8 = E_A8 (q_A8 - q_A8_us)

v_A1 = \frac{u_AA - u_A1}{R_A1} v_A2 = \frac{u_A1 - u_A2}{R_A2} v_A3 = \frac{u_A2 - u_A3}{R_A3} v_A4 = \frac{u_A3 - u_A4}{R_A4} v_A5 = \frac{u_A4 - u_A5}{R_A5} v_A6 = \frac{u_A5 - u_A6}{R_A6} v_A7 = \frac{u_A6 - u_A7}{R_A7} v_A8 = \frac{u_A7 - u_A8}{R_A8}

v_lInfPhrenicA = \frac{u_A1 - u_lAdrenalGlandC}{R_lInfPhrenicA}

\frac{d}{d t} (q_lAdrenalGlandC) = v_lInfPhrenicA - v_lAdrenalV u_lAdrenalGlandC = E_lAdrenalGlandC (q_lAdrenalGlandC - q_lAdrenalGlandC_us)

v_lAdrenalV = \frac{u_lAdrenalGlandC - u_lInfPhrenicV}{R_lAdrenalV}

\frac{d}{d t} (q_lInfPhrenicV) = v_lAdrenalV - v_lInfPhrenicV u_lInfPhrenicV = E_lInfPhrenicV (q_lInfPhrenicV - q_lInfPhrenicV_us)

v_lInfPhrenicV = \frac{u_lInfPhrenicV - u_V1}{R_lInfPhrenicV}

v_rInfPhrenicA = \frac{u_A2 - u_rAdrenalGlandC}{R_rInfPhrenicA}

\frac{d}{d t} (q_rAdrenalGlandC) = v_rInfPhrenicA - v_rAdrenalV u_rAdrenalGlandC = E_rAdrenalGlandC (q_rAdrenalGlandC - q_rAdrenalGlandC_us)

v_rAdrenalV = \frac{u_rAdrenalGlandC - u_rInfPhrenicV}{R_rAdrenalV}

\frac{d}{d t} (q_rInfPhrenicV) = v_rAdrenalV - v_rInfPhrenicV u_rInfPhrenicV = E_rInfPhrenicV (q_rInfPhrenicV - q_rInfPhrenicV_us)

v_rInfPhrenicV = \frac{u_rInfPhrenicV - u_V2}{R_rInfPhrenicV}

v_celiacA = \frac{u_A3 - u_celiacA}{R_celiacA}

\frac{d}{d t} (q_celiacA) = v_celiacA - v_liverA - v_stomachA - v_spleenA - v_pancreas1A u_celiacA = E_celiacA (q_celiacA - q_celiacA_us)

v_supMesA = \frac{u_A4 - u_supMesA}{R_supMesA}

\frac{d}{d t} (q_supMesA) = v_supMesA - v_pancreas2A - v_intestineA - v_colonA u_supMesA = E_supMesA (q_supMesA - q_supMesA_us)

v_lRenalA = \frac{u_A5 - u_lGlomerulusC}{R_lRenalA}

\frac{d}{d t} (q_lGlomerulusC) = v_lRenalA - v_lEffArteriol u_lGlomerulusC = E_lGlomerulusC (q_lGlomerulusC - q_lGlomerulusC_us)

v_lEffArteriol = \frac{u_lGlomerulusC - u_lPeritubularC}{R_lEffArteriol}

\frac{d}{d t} (q_lPeritubularC) = v_lEffArteriol - v_lRenalV u_lPeritubularC = E_lPeritubularC (q_lPeritubularC - q_lPeritubularC_us)

v_lRenalV = \frac{u_lPeritubularC - u_V4}{R_lRenalV}

v_rRenalA = \frac{u_A6 - u_rGlomerulusC}{R_rRenalA}

\frac{d}{d t} (q_rGlomerulusC) = v_rRenalA - v_rEffArteriol u_rGlomerulusC = E_rGlomerulusC (q_rGlomerulusC - q_rGlomerulusC_us)

v_rEffArteriol = \frac{u_rGlomerulusC - u_rPeritubularC}{R_rEffArteriol}

\frac{d}{d t} (q_rPeritubularC) = v_rEffArteriol - v_rRenalV u_rPeritubularC = E_rPeritubularC (q_rPeritubularC - q_rPeritubularC_us)

v_rRenalV = \frac{u_rPeritubularC - u_V5}{R_rRenalV}

v_infMesA = \frac{u_A7 - u_colonC}{R_infMesA}

v_stomachA = \frac{u_celiacA - u_stomachC}{R_stomachA}

\frac{d}{d t} (q_stomachC) = v_stomachA - v_stomachV u_stomachC = E_stomachC (q_stomachC - q_stomachC_us)

v_stomachV = \frac{u_stomachC - u_portalV}{R_stomachV}

v_spleenA = \frac{u_celiacA - u_spleenC}{R_spleenA}

\frac{d}{d t} (q_spleenC) = v_spleenA - v_spleenV u_spleenC = E_spleenC (q_spleenC - q_spleenC_us)

v_spleenV = \frac{u_spleenC - u_portalV}{R_spleenV}

v_pancreas1A = \frac{u_celiacA - u_pancreasC}{R_pancreas1A} v_pancreas2A = \frac{u_supMesA - u_pancreasC}{R_pancreas2A}

\frac{d}{d t} (q_pancreasC) = v_pancreas1A + v_pancreas2A - v_pancreasV u_pancreasC = E_pancreasC (q_pancreasC - q_pancreasC_us)

v_pancreasV = \frac{u_pancreasC - u_portalV}{R_pancreasV}

v_intestineA = \frac{u_supMesA - u_intestineC}{R_intestineA}

\frac{d}{d t} (q_intestineC) = v_intestineA - v_intestineV u_intestineC = E_intestineC (q_intestineC - q_intestineC_us)

v_intestineV = \frac{u_intestineC - u_portalV}{R_intestineV}

v_colonA = \frac{u_supMesA - u_colonC}{R_colonA}

\frac{d}{d t} (q_colonC) = v_colonA + v_infMesA - v_colonV u_colonC = E_colonC (q_colonC - q_colonC_us)

v_colonV = \frac{u_colonC - u_portalV}{R_colonV}

\frac{d}{d t} (q_portalV) = v_stomachV + v_spleenV + v_pancreasV + v_intestineV + v_colonV - v_portalV u_portalV = E_portalV (q_portalV - q_portalV_us)

v_portalV = \frac{u_portalV - u_liverC}{R_portalV}

v_liverA = \frac{u_celiacA - u_liverC}{R_liverA}

\frac{d}{d t} (q_liverC) = v_liverA + v_portalV - v_liverV u_liverC = E_liverC (q_liverC - q_liverC_us)

v_liverV = \frac{u_liverC - u_V3}{R_liverV}

\frac{d}{d t} (q_V1) = v_V2 + v_lInfPhrenicV - v_V1 \frac{d}{d t} (q_V2) = v_V3 + v_rInfPhrenicV - v_V2 \frac{d}{d t} (q_V3) = v_V4 + v_liverV - v_V3 \frac{d}{d t} (q_V4) = v_V5 + v_lRenalV - v_V4 \frac{d}{d t} (q_V5) = v_V6 + v_rRenalV - v_V5 \frac{d}{d t} (q_V6) = v_lCommonIliacV + v_rCommonIliacV - v_V6

u_V1 = E_V1 (q_V1 - q_V1_us) u_V2 = E_V2 (q_V2 - q_V2_us) u_V3 = E_V3 (q_V3 - q_V3_us) u_V4 = E_V4 (q_V4 - q_V4_us) u_V5 = E_V5 (q_V5 - q_V5_us) u_V6 = E_V6 (q_V6 - q_V6_us)

v_V1 = \frac{u_V1 - u_RA}{R_V1} v_V2 = \frac{u_V2 - u_V1}{R_V2} v_V3 = \frac{u_V3 - u_V2}{R_V3} v_V4 = \frac{u_V4 - u_V3}{R_V4} v_V5 = \frac{u_V5 - u_V4}{R_V5} v_V6 = \frac{u_V6 - u_V5}{R_V6}

v_lCommonIliacA = \frac{u_A8 - u_lLegC}{R_lCommonIliacA}

\frac{d}{d t} (q_lLegC) = v_lCommonIliacA - v_lCommonIliacV u_lLegC = E_lLegC (q_lLegC - q_lLegC_us)

v_lCommonIliacV = \frac{u_lLegC - u_V6}{R_lCommonIliacV}

v_rCommonIliacA = \frac{u_A8 - u_rLegC}{R_rCommonIliacA}

\frac{d}{d t} (q_rLegC) = v_rCommonIliacA - v_rCommonIliacV u_rLegC = E_rLegC (q_rLegC - q_rLegC_us)

v_rCommonIliacV = \frac{u_rLegC - u_V6}{R_rCommonIliacV}

q_TotHeart = q_RA + q_RV + q_LA + q_LV q_TotLung = q_lungC + q_pulmA q_TotHead = q_brachiocephalicA + q_rSubclavianA + q_lSubclavianA + q_headC q_TotThoracic = q_lAdrenalGlandC + q_lInfPhrenicV + q_rAdrenalGlandC + q_rInfPhrenicV q_TotAorta = q_AA + q_A1 + q_A2 + q_A3 + q_A4 + q_A5 + q_A6 + q_A7 + q_A8 q_TotRenal = q_lGlomerulusC + q_lPeritubularC + q_rGlomerulusC + q_rPeritubularC q_TotGut = q_celiacA + q_supMesA + q_stomachC + q_spleenC + q_pancreasC + q_intestineC + q_colonC + q_portalV + q_liverC q_TotVenaCava = q_V1 + q_V2 + q_V3 + q_V4 + q_V5 + q_V6 q_TotLimbs = q_lArmC + q_rArmC + q_lLegC + q_rLegC q_TotBlood = q_TotHeart + q_TotLung + + q_TotHead + q_TotThoracic + q_TotAorta + q_TotRenal + q_TotGut + q_TotVenaCava + q_TotLimbs