Model Mathematics

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Component: Circ_Environment

hrf = \frac{HR}{0.06}

Component: LATiming

beattime = \frac{realtime - (⌊ \frac{realtime}{hrf} ⌋ hrf)}{1000}

Component: LAElastanceFunction

Ts = TsK \sqrt{\frac{hrf}{1000}} E_LA = \{\begin{matrix} Edia + \frac{(Esys - Edia) (1 - (\cos (\frac{π time}{Ts})))}{2} if time \geq 0 \land time ≦ Ts \\ Edia + \frac{(Esys - Edia) (1 + \cos (\frac{2 π (time - Ts)}{Ts}))}{2} if time < 1.5 Ts \land time \geq Ts \\ Edia otherwise \end{matrix}

Component: LeftAtrium

\frac{d}{d time} (sub_X_LA) = \frac{conc_X_PV F_LA - (conc_X_LA F_LV)}{1000} conc_X_LA = \frac{sub_X_LA}{V_LA} P_LA = E_LA (V_LA - V_LA_0) + P_LA_ext F_LA = \frac{P_PV - P_LA}{R_LA} \frac{d}{d time} (V_LA) = \frac{F_LA - F_LV}{1000}

Component: LVTiming

beattime = \frac{realtime - (⌊ \frac{realtime}{hrf} ⌋ hrf)}{1000} - PRint

Component: LVElastanceFunction

Ts = TsK \sqrt{\frac{hrf}{1000}} E_LV = \{\begin{matrix} Edia + \frac{(Esys - Edia) (1 - (\cos (\frac{π time}{Ts})))}{2} if time \geq 0 \land time ≦ Ts \\ Edia + \frac{(Esys - Edia) (1 + \cos (\frac{2 π (time - Ts)}{Ts}))}{2} if time < 1.5 Ts \land time \geq Ts \\ Edia otherwise \end{matrix}

Component: LeftVentricle

fiber_stress = P_LV (1 + \frac{3 V_LV}{200}) \frac{d}{d time} (sub_X_LV) = \frac{conc_X_LA F_LV - (conc_X_LV (F_AO + F_CO))}{1000} conc_X_LV = \frac{sub_X_LV}{V_LV} P_LV = E_LV (V_LV - V_LV_0) + pLVPext F_LV = \{\begin{matrix} \frac{P_LA - P_LV}{R_LV} if P_LA > P_LV \\ 0 otherwise \end{matrix} \frac{d}{d time} (V_LV) = \frac{F_LV - F_AO - F_CO}{1000}

Component: Aorta

\frac{d}{d time} (sub_X_AO) = \frac{conc_X_LV F_AO - (conc_X_AO (F_CR + F_AR))}{1000} conc_X_AO = \frac{sub_X_AO}{V_AO} \frac{d}{d time} (F_AOalways) = \frac{\frac{1}{L_AO} (P_LV - P_AO) - (\frac{R_AO}{L_AO} F_AOalways)}{1000} F_AO = \{\begin{matrix} F_AOalways if P_LV > P_AO \\ 0 otherwise \end{matrix} \frac{d}{d time} (V_AO) = \frac{F_AO - F_AR - F_CR}{1000} P_AO = \frac{1}{C_AO} (V_AO - V_AO_0)

Component: CoronaryCirc

\frac{d}{d time} (sub_X_CO) = \frac{conc_X_LV F_CO - (conc_X_CO F_COV)}{1000} conc_X_CO = \frac{sub_X_CO}{V_CO} \frac{d}{d time} (V_CO) = \frac{F_CO - F_COV}{1000} P_CO = \frac{1}{C_CO} (V_CO - V_CO_0) F_CO = \{\begin{matrix} \frac{P_LV - P_CO}{R_CO} if P_LV > P_CO \\ 0 otherwise \end{matrix} F_COV = \frac{P_CO - P_RA}{R_COV}

Component: CarotidCirc

conc_X_CRtissue = \frac{sub_X_CRtissue}{V_CRtissue} \frac{d}{d time} (sub_X_CRtissue) = \frac{(conc_X_CR - conc_X_CRtissue) D_CRtissue}{1000} \frac{d}{d time} (sub_X_CR) = \frac{conc_X_AO F_CR - (conc_X_CR F_CRV) - ((conc_X_CR - conc_X_CRtissue) D_CRtissue)}{1000} conc_X_CR = \frac{sub_X_CR}{V_CR} \frac{d}{d time} (V_CR) = \frac{F_CR - F_CRV}{1000} P_CR = \frac{1}{C_CR} (V_CR - V_CR_0) F_CR = \frac{P_AO - P_CR}{R_CR} F_CRV = \frac{P_CR - P_VC}{R_CRV}

Component: Arteries

\frac{d}{d time} (sub_X_AR) = \frac{conc_X_AO F_AR - (conc_X_AR (F_SK + F_AD + F_MU + F_GI + F_LI + F_KI + F_OT))}{1000} conc_X_AR = \frac{sub_X_AR}{V_AR} \frac{d}{d time} (V_AR) = \frac{F_AR - F_AD - F_MU - F_GI - F_LI - F_KI - F_OT - F_SK}{1000} P_AR = \frac{1}{C_AR} (V_AR - V_AR_0) F_AR = \frac{P_AO - P_AR}{R_AR}

Component: Adipose

\frac{d}{d time} (sub_X_AD) = \frac{conc_X_AR F_AD - (conc_X_AD F_ADV)}{1000} conc_X_AD = \frac{sub_X_AD}{V_AD} \frac{d}{d time} (V_AD) = \frac{F_AD - F_ADV}{1000} F_ADV = \frac{P_AD - P_VE}{R_ADV} P_AD = \frac{1}{C_AD} (V_AD - V_AD_0) F_AD = \frac{P_AR - P_AD}{R_AD}

Component: Muscle

\frac{d}{d time} (sub_X_MU) = \frac{conc_X_AR F_MU - (conc_X_MU F_MUV)}{1000} conc_X_MU = \frac{sub_X_MU}{V_MU} \frac{d}{d time} (V_MU) = \frac{F_MU - F_MUV}{1000} F_MUV = \frac{P_MU - P_VE}{R_MUV} P_MU = \frac{1}{C_MU} (V_MU - V_MU_0) F_MU = \frac{P_AR - P_MU}{R_MU}

Component: GutIntestine

\frac{d}{d time} (sub_X_GI) = \frac{conc_X_AR F_GI - (conc_X_GI F_GIV)}{1000} conc_X_GI = \frac{sub_X_GI}{V_GI} \frac{d}{d time} (V_GI) = \frac{F_GI - F_GIV}{1000} F_GIV = \frac{P_GI - P_LI}{R_GIV} P_GI = \frac{1}{C_GI} (V_GI - V_GI_0) F_GI = \frac{P_AR - P_GI}{R_GI}

Component: Liver

\frac{d}{d time} (sub_X_LI) = \frac{(conc_X_GI F_GIV + conc_X_AR F_LI - (conc_X_LI F_LIV)) + conc_X_LI metabolism}{1000} conc_X_LI = \frac{sub_X_LI}{V_LI} \frac{d}{d time} (V_LI) = \frac{F_GIV + F_LI - F_LIV}{1000} F_LIV = \frac{P_LI - P_VE}{R_LIV} P_LI = \frac{1}{C_LI} (V_LI - V_LI_0) F_LI = \frac{P_AR - P_LI}{R_LI}

Component: Kidney

\frac{d}{d time} (sub_X_KI) = \frac{conc_X_AR F_KI - (conc_X_KI F_KIV)}{1000} conc_X_KI = \frac{sub_X_KI}{V_KI} \frac{d}{d time} (V_KI) = \frac{F_KI - F_KIV}{1000} F_KIV = \frac{P_KI - P_VE}{R_KIV} P_KI = \frac{1}{C_KI} (V_KI - V_KI_0) F_KI = \frac{P_AR - P_KI}{R_KI}

Component: Skin

\frac{d}{d time} (sub_X_SK) = \frac{conc_X_AR F_SK - (conc_X_SK F_SKV)}{1000} conc_X_SK = \frac{sub_X_SK}{V_SK} \frac{d}{d time} (V_SK) = \frac{F_SK - F_SKV}{1000} F_SKV = \frac{P_SK - P_VE}{R_SKV} P_SK = \frac{1}{C_SK} (V_SK - V_SK_0) F_SK = \frac{P_AR - P_SK}{R_SK}

Component: OtherTissue

\frac{d}{d time} (sub_X_OT) = \frac{conc_X_AR F_OT - (conc_X_OT F_OTV)}{1000} conc_X_OT = \frac{sub_X_OT}{V_OT} \frac{d}{d time} (V_OT) = \frac{F_OT - F_OTV}{1000} F_OTV = \frac{P_OT - P_VE}{R_OTV} P_OT = \frac{1}{C_OT} (V_OT - V_OT_0) F_OT = \frac{P_AR - P_OT}{R_OT}

Component: Veins

\frac{d}{d time} (sub_X_VE) = \frac{conc_X_SK F_SKV + conc_X_AD F_ADV + conc_X_MU F_MUV + conc_X_LI F_LIV + conc_X_KI F_KIV + conc_X_OT F_OTV - (conc_X_VE F_VEV)}{1000} conc_X_VE = \frac{sub_X_VE}{V_VE} F_VE = F_SKV + F_ADV + F_MUV + F_LIV + F_KIV + F_OTV \frac{d}{d time} (V_VE) = \frac{F_VE - F_VEV}{1000} P_VE = \frac{1}{C_VE} (V_VE - V_VE_0) F_VEV = \frac{P_VE - P_VC}{R_VEV}

Component: VenaCava

\frac{d}{d time} (sub_X_VC) = \frac{conc_X_VE F_VEV + conc_X_CR F_CRV - (conc_X_VC F_VCV)}{1000} conc_X_VC = \frac{sub_X_VC}{V_VC} F_VC = F_CRV + F_VEV F_VCV = \frac{P_VC - P_RA}{R_VCV} \frac{d}{d time} (V_VC) = \frac{F_VC - F_VCV}{1000} P_VC = \frac{1}{C_VC} (V_VC - V_VC_0)

Component: RATiming

beattime = \frac{realtime - (⌊ \frac{realtime}{hrf} ⌋ hrf)}{1000}

Component: RAElastanceFunction

Ts = TsK \sqrt{\frac{hrf}{1000}} E_RA = \{\begin{matrix} Edia + \frac{(Esys - Edia) (1 - (\cos (\frac{π time}{Ts})))}{2} if time \geq 0 \land time ≦ Ts \\ Edia + \frac{(Esys - Edia) (1 + \cos (\frac{2 π (time - Ts)}{Ts}))}{2} if time < 1.5 Ts \land time \geq Ts \\ Edia otherwise \end{matrix}

Component: RightAtrium

\frac{d}{d time} (sub_X_RA) = \frac{conc_X_VC F_VCV + conc_X_CO F_COV - (conc_X_RA F_RV)}{1000} conc_X_RA = \frac{sub_X_RA}{V_RA} P_RA = E_RA (V_RA - V_RA_0) + P_RA_ext F_RA = F_VCV + F_COV \frac{d}{d time} (V_RA) = \frac{F_RA - F_RV}{1000}

Component: RVTiming

beattime = \frac{realtime - (⌊ \frac{realtime}{hrf} ⌋ hrf)}{1000} - PRinterval

Component: RVElastanceFunction

Ts = TsK \sqrt{\frac{hrf}{1000}} E_RV = \{\begin{matrix} Edia + \frac{(Esys - Edia) (1 - (\cos (\frac{π time}{Ts})))}{2} if time \geq 0 \land time ≦ Ts \\ Edia + \frac{(Esys - Edia) (1 + \cos (\frac{2 π (time - Ts)}{Ts}))}{2} if time < 1.5 Ts \land time \geq Ts \\ Edia otherwise \end{matrix}

Component: RightVentricle

\frac{d}{d time} (sub_X_RV) = \frac{conc_X_RA F_RV - (conc_X_RV F_PA)}{1000} conc_X_RV = \frac{sub_X_RV}{V_RV} P_RV = E_RV (V_RV - V_RV_0) + P_RV_ext F_RV = \{\begin{matrix} \frac{P_RA - P_RV}{R_RV} if P_RA > P_RV \\ 0 otherwise \end{matrix} \frac{d}{d time} (V_RV) = \frac{F_RV - F_PA}{1000}

Component: PulmonaryArtery

\frac{d}{d time} (sub_X_PA) = \frac{conc_X_RV F_PA - (conc_X_PA F_Pa)}{1000} conc_X_PA = \frac{sub_X_PA}{V_PA} F_PA = \{\begin{matrix} \frac{P_RV - P_PA}{R_PA} if P_RV > P_PA \\ 0 otherwise \end{matrix} \frac{d}{d time} (V_PA) = \frac{F_PA - F_Pa}{1000} P_PA = \frac{1}{C_PA} (V_PA - V_PA_0)

Component: PulmonaryArteries

\frac{d}{d time} (sub_X_Pa) = \frac{conc_X_PA F_Pa - (conc_X_Pa (F_PC + F_SH))}{1000} conc_X_Pa = \frac{sub_X_Pa}{V_Pa} \frac{d}{d time} (V_Pa) = \frac{F_Pa - F_PC - F_SH}{1000} P_Pa = \frac{1}{C_Pa} (V_Pa - V_Pa_0) F_Pa = \frac{P_PA - P_Pa}{R_Pa}

Component: Shunt

F_SH = \frac{P_Pa - P_PC}{R_SH}

Component: PulmonaryCapillaries

\frac{d}{d time} (sub_X_PC) = \frac{conc_X_Pa F_PC - (conc_X_PC F_PCV)}{1000} conc_X_PC = \frac{sub_X_PC}{V_PC} \frac{d}{d time} (V_PC) = \frac{F_PC - F_PCV}{1000} P_PC = \frac{1}{C_PC} (V_PC - V_PC_0) F_PC = \frac{P_Pa - P_PC}{R_PC} F_PCV = \frac{P_PC - P_PV}{R_PCV}

Component: PulmonaryVein

\frac{d}{d time} (sub_X_PV) = \frac{conc_X_PC F_PCV + conc_X_Pa F_SH - (conc_X_PV F_LA)}{1000} conc_X_PV = \frac{sub_X_PV}{V_PV} F_PV = F_PCV + F_SH \frac{d}{d time} (V_PV) = \frac{F_PV - F_LA}{1000} P_PV = \frac{1}{C_PV} (V_PV - V_PV_0)