12.31.2003 Replaced Fav in systemic model to Flv (from heart model). Changed Vra:t = Fvc - Fra to Vra:t = Fvc - Fra + Fcrb + Fcor; under heart section to include cerebral, coronary flow return to ra.-MN 01.13.2004 Changed model so distal end of cerebral blood network empties into proximal vena cavae. Fcrb = (Paop-Pvc)/Rcrb; ("Equations of systemic circulation"/"Flows") Previously Fcrb = (Paop-Pra)/Rcrb; Ftvc = Fsv + Fcrb - Fvc; ("Equations of systemic circulation"/"Transmural Flows") Previously Ftvc = Fsv - Fvc; Vra:t = Fvc - Fra + Fcor; ("Eqautions of varying elastance heart model"/"Right Atrium and its Contraction") Previously Vra:t = Fvc - Fra + Fcrb + Fcor; -MN 01.28.2004 Added a variable to calculate cardiac output (Vt). -MN 02.10.2004 Ran model using max8 par set from sbheart.proj. t.max=10 and t.delta = 0.01. Radau solver Inf or NaN at ~t=2. Ran model with same par set. t.max = 10 and t.delta=0.005. Modle run did not produce a visible time-elapsed bar. (Did not "turn over"). Ran model again with same par set. t.max=10 and t.delta=0.005. Model woulnd't turn over. Ran model with same par set t.max = 10 and t.delta=0.01. Model run wouldn't turn over! ??????????? - Big report sent to EB. Added if statements to baroreceptor equations to avoid NaNs due to time delay. For example: T_hrv*(N_hrv:t) + N_hrv = K_hrv*Nbr(t-L_hrv); became T_hrv*(N_hrv:t) + N_hrv = if (t<L_hrv) 0 else (K_hrv*Nbr(t-L_hrv)); Model now finishes run without Inf/NaN error using max8 parset t.max=10 and t.delta = 0.01 (same conditions as first note under today's heading). But only if a = 0.01. If a returned to 0.001, model blows up with these parameters. You have to change t.delta (0.001 sec worked) in order to run the model successfully. -MN 02.12.2004 Model completes run with max8 parset, t.max=10, t.delta = 0.002. Blows up with same conditions and 0.003 time step. Blows at 23.16 sec if run for 25 sec using t.delta=0.002. Blows at 22.693 sed if run for 25 sec using t.delta=0.001.-MN 02.13.2004 Does not blow for run of 25 sec with t.delta=0.0005. However, JSim runs out of memory when trying to look at data file for plotpage 1. Model seems to reach a steady state near 25 sec. 02.16.2004 Change parameter "a" to 0.01, runs successfully at 0.001 s timestep for 25 sec. Runs successfully with a = 0.01, t.delta = 0.01 for 25 although it looks like it might blow after 25 sec. Runs successfully with a =0.01, t.delta = 0.01 for 35 seconds (computing time = 13 minutes). Blows up arond 8 seconds when run as directly above, but with a 0.1 sec time step. If a=0.1 model runs succesfully for 35 sec with t.delta=0.01, but develops differently than previous successful run, although steady state values may be similar. Run took about 7.5 minutes. Blows at 25.26 sec if run for 60 sec with a=0.01, t.delta=0.01, max8 parset. Blows at 22.6 sec if run for 35 sec with a=0.01, t.delta=0.01, ????????????????????? (had successful run before) Tried this again and it blew again at same point. 02.23.2004 Changed efferent pathway code to account for time delay. T_x*(N_x:t) + N_x = if (t>L_x) (K_x*Nbr(t-L_x)) else (K_x*Nbr(t.min)); became: N_x:t = if (t<L_x) (-N_x + (K_x*Nbr(t-L_x)))/T_x) else 0; Integrated contractility control by coding Lu et al. eq. 9, 11, 12. Ran successfully with parset max8 (t.max = 10 sec, t.delta=0.01 sec, a = 0.001 sec) Some parameters for efferent baro pathways not correct in code (mostly from Lu et al. Table 6). They were corrected. Ran for 10 sec with 0.005 time step (a=0.001), blew up after t=6 sec, but pressures and HR look more like they did in archived spheart.proj file. 02.24.2004 Blows just before t=1 sec for a run of 7 sec with t.delta=0.01. Successful run for 10 sec with 0.001 time step (a=0.001). Run took 6 minutes. Blows just after t=1 sec if t.max=7, t.delta=0.01, K=0.75, a=0.001. Successful run of 7 sec for t.delta = 0.005 with K=0.75. Unsuccessul run of 10 sec for t.delta = 0.005 with K=0.75 (blows around t = 9 sec.) Successful run of 10 sec for t.delta = 0.005 with K=0.5. Successful run of 10 sec for t.delta = 0.0025 with K=1. Run took 6'20". Successful run of 10 sec for t.delta = 0.004 (K=1, a=0.001). Run took 5'53". System appears to become unstable toward end of run. Unsuccessful run of 20 sec at t.delta = 0.0025. Run took 14'. Blew up at t=17.0525 sec. Successful run of 20 sec at t.delta = 0.001. Run took 19'20". 02.25.2004 Successful run of 20 sec. Max8 parset. t.delta=0.001, a2=0.018. Run took 6'30". Unsuccessful run of 20 sec. max8parset. t.delta=0.01, a2=0.018. Blows up around 14 sec. Successful run of 20 sec. max8 parset. t.delta=0.005, a2=0.018. Run took 5'54". Successful run of 20 sec. max8 parset. t.delta=0.005, a2=0.18. Run took 3'. SUccessful run of 20 sec. max8 parset. t.delta=0.01, a2=0.18. Run took 6'50". Appears unstable, however, towards end of run. Successful run of 20 sec. max8 parset. t.delta=0.01, a2=0.18, a=0.1. Run took 26". Appears stable. Successful run of 20 sec. max8 parset. t.delta=0.05, a2=0.18, a=0.1. Run took 22". Appears stable. Successful run for 30 sec. t.delta=0.01, a2=0.18, a=0.01. Run took 2'. Unsuccessful run for 30 sec. t.delta=0.01, a2=0.18, a=0.001. Blew around 24 sec. Using fgen_1. fgen_1 is a pulse 1 function: 50 millilters of blood loss over 5 seconds beginning at t=13 sec. Successful run for 30 sec. t.delta=0.005, a2=0.18, a=0.001. fgen_1 used. Appears unstable from 26-30 sec. Run took 7'. Successful run for 30 sec. t.delta=0.005, a2=0.18, a=0.001. Fx=0. Appears unstable from 26-30 sec. Run took 7'. Successful run for 30 sec. t.delta=0.01. a2=0.018, a=0.01. Run took 3'15". Pressures look different (lower) than preceding runs. What makes pressure lower??? 03.04.2004 Successful run for 30 sec. t.delta = 0.0005. max8 parset. Run took 70'. Unsuccesful run of 30 sec. t.delta=0.01. a=0.001 a2=0.018. Becomes unstable at 12 sec. Blows at 28 sec. TABLE OF SUCCESSFUL RUNS (model doesn't blow up) (other parameters as in parset "max8") t.delta a a2 t.max t.min duration fxn gen used? ---------------------------------------------------------------------- 0.0005" 0.001 0.0018 30" 0" 70' NO 0.01" 0.01 0.018 30" 0" 3'15" NO 0.01" 0.01 0.018 30" 0" 3'12" YES 0.005" 0.001 0.18 30" 0" 7' NO 0.005" 0.001 0.18 30" 0" 7' YES 0.01" 0.01 0.18 30" 0" 2' NO 0.05" 0.1 0.18 20" 0" 22" NO 0.01" 0.1 0.18 20" 0" 26" NO 0.005" 0.001 0.0018 30" 0" 16.5' NO unstable Performance problems resolved by using RK4 solver. Works fast for default baroreceptor time constants. 03.17.2004 Coded in a function to keep heart rate constant during atrial or ventricular contraction. Replaced the HR in y and ya equations with a follower variable called HR2 that doesn't change when y or ya or yx or yxa is above its cutoff value. -MN 03.24.2004 Coded in a function to keep contractility constant during atrial or ventricular contraction. Replaced the "aF_con" term in the systolic elastance equation with a follower variable "aF_con2" which does not change when y or ya or yx or yxa is above its cutoff value. For a run of 60 seconds (using a single parameter set that gives normal physiology) this model appears much less stable without the functions that keep heartrate and contractility from changing during contraction. Without this additional control, the heartbeat becomes highly irregular with many interrupted beats (even though heartrate remains within a normal range) and there is a much higher variation in contraction time. -MN 03.29.2004 Cleaned up code and project file, finished code markup. Archived model as sbpheart.proj. -MN 04.062004 Began integration of pulmonary mechanics model written by Melissa Kreuger. Melissa's notes: THIS MODEL IS BASED ON ATHANASIADES ET AL. ENERGY ANALYSIS OF A NONLINEAR MODEL OF THE NORMAL HUMAN LUNG. J BIOL SYS. 8(2):115-39, 2000 THIS MODEL IS A LUMPED PARAMETER DESCRIPTION OF LUNG MECHANICS. THE MODEL CONSIDERS AIRWAY RESISTANCE IN THREE COMPARTMENTS: SMALL, COLLAPSIBLE, AND UPPER AIRWAYS. EACH OF THESE COMPARTMENTS HAS A RESISTANCE DEPENDENT ON EITHER COMPARTMENTAL VOLUME OR FLOW. THERE ARE ALSO SEVERAL COMPLIANT COMPARTMENTS: THE COLLAPSIBLE AIRWAYS, LUNG PARENCHYMA, AND CHEST WALL. THE MODEL IS DRIVEN BY A MUSCLE PRESSURE. WHILE THE MODEL LOOKS TO BE VERY STRAIGHT-FORWARD THE MANUSCRIPT IS PLAGUED WITH SMALL ERRORS THAT MAKE IMPLEMENTATION DIFFICULT. I WILL TRY TO MARK THE DIFFERENCES BETWEEN THIS CODE AND THEIR MODEL WHERE APPROPRIATE BELOW. Melissa Krueger Pulmonary & Critical Care Medicine University of Washington Feb 25, 2004 -MN Integration went smoothly. Markup completed. Model archived as spbha.proj. -MN 04.13.2004 Using the new "event" function available in JSim, I changed the semi-dynamic heart rate and contractility variables to discrete functions. Heart rate and contractility values used in the model equations are now constant across the entire cardiac cycle. They update at a single timepoint - the first timepoint after a full cycle has been completed. The markup was changed to account for the new characteristics of these variables. The model was archived as spbha.proj. The CVS repository should show this as the second version of spbha.proj. -MN 04.14.2004 For next version of VS_HIP: 1) tissue/lung gas exchange 2) chemosensitivity 3) influence of pleural pressure on heart, VC Introduced influence of pleural pressure on heart and vena cava. New variables: Praf, Plaf, Plvf, Prvf (internal pressures of heart chambers generated by the contracting free walls); Pvci (internal pressure of vena cava) Note: now Lu et al. P-V relationship (Eqs. 2 and 3) for vena cava is for internal pressure, not transmural pressure (some parameters in that equation may need to be adjusted, but i'm not sure). Corrected Paop equation from Paop1 = (-Rtaop*Caop*Rcor*Rav*Rcrb*Faop+Vaop*Rcor*Rav*Rcrb+ Rtaop*Rav*Rcrb*Caop*(((((((Pra)))))))+Rtaop*Rav*Rcor*Caop*Pra+Rtaop*Rcor*Rcrb*Caop*Plv)/ (Caop*(Rcor*Rav*Rcrb+Rav*Rcrb*Rtaop+Rcor*Rcrb*Rtaop+Rav*Rcor*Rtaop));// Eqs. B,C,D to Paop1 = (-Rtaop*Caop*Rcor*Rav*Rcrb*Faop+Vaop*Rcor*Rav*Rcrb+ Rtaop*Rav*Rcrb*Caop*(((((((Pvc)))))))+Rtaop*Rav*Rcor*Caop*Pra+Rtaop*Rcor*Rcrb*Caop*Plv)/ (Caop*(Rcor*Rav*Rcrb+Rav*Rcrb*Rtaop+Rcor*Rcrb*Rtaop+Rav*Rcor*Rtaop));// Eqs. B,C,D because cerebral criculation ends at proximal vena cava, not right atrium. Paop2 = (Rcor*Rcrb*Vaop-Rtaop*Caop*Rcor*Rcrb*Faop+ Rtaop*Caop*Rcrb*Pra+Rtaop*Caop*Rcor*(((((((Pra))))))))/ (Rcor*Rcrb+Rcrb*Rtaop+Rtaop*Rcor)/Caop; changed to Paop2 = (Rcor*Rcrb*Vaop-Rtaop*Caop*Rcor*Rcrb*Faop+ Rtaop*Caop*Rcrb*Pra+Rtaop*Caop*Rcor*(((((((Pvc))))))))/ (Rcor*Rcrb+Rcrb*Rtaop+Rtaop*Rcor)/Caop; for same reason. Vc:t = (Pmus-Pc-Pcw)/(Ru+Rs) - (Pc-Pl-Pve)/Rs; // shouldn't this be Vc:t = (Pmus-Pc-Pcw)/(Ru+Rc) - (Pc-Pl-Pve)/Rs; ? -MN 04.29.2004 Finished implementing gas exchange model from K. Vinnakota. Went back to original systemic models to tweak system such that there is sufficient flow in pul and sys capillaries (~83 ml/s). Created a new tuned parameter set that gives pretty good textbook physiology (aortic waveform will need some work). Tried to run it with pleural pressure influencing heart chamber and vena cava pressure, but it appears a significant amount of fine tuning will be necessary once pleural influence is instated. 04.30.2004 Took out pleural pressure influence (see 04.14.2004). Began to finish markup for this model which will be archived as spbhag.proj. 05.03.2004 Finished cleaning up code for archival. Archived as spbhag.proj in "archived model" folder and as "VS_HIP" for CVS archive. The name "VS_HIP" will be given to the most advanced HIP model developed for the virtual soldier project that is archived within CVS. As advances are made, there will be concurrent versions of the "VS_HIP" model. -MN It appears that Lu et al (2003) did not use the Athanasiades airway mechanics model in conjunction with their chemoreceptor model. Looks like they used the airway model described in Liu et al. (1998), where Ppl is the driving functions, not Pmus. This model was originally published in the 1970s and the Atahnasiades model appears to be more descriptive. Can Ppl be made into the driving function for our airway model? 05.06.2004 Put Ppl influence back into model, created text physio. parameter set. Reduced Rsv and Eminlv. Worked on incorporating o2co2_v3 blood gas handling model. Tripped up by /g units in o2co2_v3 model, they aren't in vinnakota model (he took out the per gram units by multiplying by grams of tissue. grams of tissue assumed to be 60 kg.) Eventually i got something that seems to be doing grossly what it should. need to look up textbook bicarb, [H+], etc. values for blood. pH seems to be off quite a bit, but some parameter sets in the original o2co2_v3 model deliver OK values. will need to do some tuning for [H+] values. Peripheral chemoreceptors were also incorporated into this model. basis for this: Lu et al. (2003) -MN 05.07.2004 Parameters DA, DB and DC were changed so that the Ppl function would give values similar to the Athanasiades model Ppl function when the initial gc value (0.087) was plugged in to the Ppl chemoreceptor equations. 05.11.2004 Acid rate excretion in man = 1.5 mEq/hour = 4.17E-7 mol/sec for 5 L of blood recirculating: (Ruch 1974 p. 512). 1 mEq = 1 mmol. For Visf = 48 L, excretion rate = 8.69E-9 M/sec. 05.20.2004 Parameters KA, KB, KC, DA, DB and DC were tuned so that model gives results similar to the chemoreflex studies of Duffin et al. (2000) during hypoxia and hypercapnia. -MN 05.24.2004 Changed discontinuous heartrate function event to event (t+t.delta>=(tshift+(1/HR2))) { ... it was event (t>=(tshift+(1/HR2))) { ... this corrected irregularities in the initial increase of the systolic elastance curve. (sequencing issue?) -MN 05.25.2004 Threw in 3 switches to 1) make baroreceptor influence constant (all normalized firing frequencies are 0.5), 2) remove pleural pressure influence on the circulation and 3) remove the influence of the pericardium. The codenames, respectively are: BARO1on0off PPL1on0off PERI1on0off -MN 06.01.2004 Average weight of adult human heart = 310 g. Ref: http://sln.fi.edu/biosci/structure/structure.html Heart volume is about 1/10 total blood volume in body: Rushmer. CARDIOVASCULAR DYNAMICS. p. 61. // TITLE: spbhagt // Downloaded May 4, 2005 and renamed CVRESP // SUBTITLE: // A closed loop cardiopulmonary model composed of a four-chamber // varying-elastance heart, a systemic circulation, a // pulmonary circulation, airways mechanics, baroreceptors, gas // exchange and blood gas handling. // The systemic and pulmonary circulations were adapted from // earlier models coded by Dan Beard: "syscirc" and "pulcirc." // Dan Beard also coded the baroreception. All three of these // models are based on the Lu et al. (2001) publication. // The varying elastance heart model was coded by Jim Bassingthwaighte. // Melissa Krueger coded the airways mechanics component, which is // based on the model by Athanasiades et al. (2000) Kalyan Vinnakota // coded the gas exchange model, which is based on Lu et al. (2001). // Ranjan K. Dash coded the blood gas handling model, which employs // concepts from mass transport and acid-base kinetics (see Woodbury // 1974). // An external variable named "Fx" is included to represent blood // lost from the left ventricle, and a variable named "Hin" represents // acid infused into the systemic circulation. // MODEL ASSEMBLED BY: // // Maxwell Neal // National Simulation Resource // Department of Bioengineering // Box 357962 // University of Washington // Seattle, WA 98195 // USA // mneal@nsr.bioeng.washington.edu // CONTENTS: // _____________________________________________________________________________ // * Operations manual // // * Inputs // // * Outputs // // * Code // // I. Parameters of varying elastance heart model // II. Parameters of systemic circulation // III. Parameters of pulmonary circulation // IV. Parameters of baroreceptor model // - IVa. Vagal heart rate pathway // - IVb. Sympathetic heart rate pathway // - IVc. Sympathetic contractility pathway // - IVd. Sympathetic vasomotor tone pathway // - IVe. Heart rate control parameters // - IVf. Contractility control parameters // V. Parameters of airway mechanics // VI. Parameters of gas exchange // VII. Parameters of blood gas handling // VIII. Variables of varying elastance heart model // IX. Variables of systemic circulation // X. Variables of pulmonary circulation // XI. Variables of baroreceptor model // XII. Variables of airway mechanics // XIII. Variables of gas exchange // XIV. Variables of blood gas handling // XV. Equations of varying elastance heart model // XVI. Equations of systemic circulation // XVII. Equations of pulmonary circulation // XVIII. Equations of baroreceptor model // XIX. Equations of airway mechanics // XX. Equations of gas exchange // XXI. Equations of blood gas handling // // * Assumptions & Limitations // // * Comments // // * Parameter and variable glossary // // * Schematics // // * Pulblished parameter values // // * References // _____________________________________________________________________________ // OPERATIONS MANUAL: // JSim users: Import model into new project and compile. // The user may alter parameter values in the Run Time // dialog to visualize effects to the system. Default // values are from the tuned parameter set supplied in the // JSim project file. Original parameter values from the // literature are listed under "Published parameter values." // SOLVER USED FOR TUNED PARAMETER SET: RK4, step size = 20 // External function values: // Pao=760 mmHg // r_Pao_O2=0.21 // r_Pao_CO2=0.0003 // Fx=0 ml/sec // Hin=0 M/sec // INPUTS: // CODEWORD DESCRIPTION //------------------------------------------------------------------------------ // f Breathing frequency (breaths/min) // FRC Minimum lung volume // H Hematocrit // Pao External atmospheric pressure // Print Difference in atrial and ventricular activation times // RQ Respiratory quotient // RV Residual volume // Tbody Body temperature // tweight Tissue weight of body // TLC Total lung capacity // VD Lung dead space // Visf Systemic interstitial fluid volume // OUTPUTS: // CODEWORD DESCRIPTION //------------------------------------------------------------------------------ // afs_con Dynamic heart contractility scaling function // afs_con2 Discrete heart contractility scaling function // B_isf HCO3- ION CONC in SYST ISF // B_pc HCO3- ION CONC in PULM CAP // B_sc HCO3- ION CONC in SYST CAP // CO2flux CO2 flux from alveoli to pulmonary capillaries // CtCO2_pc Concentration of CO2 in pulmonary capillaries // CtCO2_sc Concentration of CO2 in systemic capillaries // CtO2_pc Concentration of O2 in pulmonary capillaries // CtO2_sc Concentration of O2 in systemic capillaries // DL_CO2 Diffusing capacity of lungs to CO2 // DL_O2 Diffusing capacity of lungs to O2 // dV Chest wall volume follower to get flow // Ela Time-varying elastance of left atrium // Elv Time-varying elastance of left ventricle // Era Time-varying elastance of right atrium // Erv Time-varying elastance of right ventricle // Faod Distal aorta flow // Faop Proximal aorta flow // Fcor Coronary flow // Fcrb Cerebral flow // F_con Normalized discharge rate for contractility control // F_hrs Normalized discharge rate for sympathetic heart rate control // F_hrv Normalized vagal discharge rate // F_vaso Normalized discharge rate for vasomotor control // Fla Flow across mitral valve // Flv Flow across aortic valve // Fpa Pulmonary arterioles flow // Fpad Distal pulmonary arterial flow // Fpap Proximal pulmonary arterial flow // Fpc Pulmonary capillaries flow // Fps Pulmonary shunt flow // Fpv Pulmonary veins flow // Fra Flow across tricuspid valve // Frv Flow across pulmonary valve // Fsa Systemic arteries flow // Fsad Systemic arterioles flow // Fsc Systemic capillaries flow // Fsv Systemic veins flow // Ftaod Distal aorta radial flow // Ftaop Proximal aorta radial flow // Ftpa Pulmonary arterioles radial flow // Ftpad Distal pulmonary arterial radial flow // Ftpap Proximal pulmonary arterial radial flow // Ftpc Pulmonary capillaries radial flow // Ftpv Pulmonary veins radial flow // Ftsa Systemic arteries radial flow // Ftsad Systemic arterioles radial flow // Ftsc Systemic capillaries radial flow // Ftsv Systemic veins radial flow // Ftvc Vena cava radial flow // Fvc Vena cava flow // GO2isf GULOSITY FOR O2 CONSUMPTION in TISSUE // H_isf H+ ION CONC in SYST ISF // H_pc H+ ION CONC in PULM CAP // H_sc H+ ION CONC in SYST CAP // Hout Variable for H+ ion sink (crude kidney) // HR Dynamic heart rate // HR2 Discrete heart rate // N_con Sympathetic discharge rate at CNS for contractility // N_hrs Sympathetic discharge rate at CNS controlling heart rate // N_hrv Vagal discharge rate at CNS controlling heart rate // N_vaso Sympathetic discharge rate at CNS controlling vasomotor tone // Nbr Instantaneous firing frequency of baroreceptor // O2flux O2 flux from alveoli to pulmonary // PA Alveolar pressure // PA_CO2 Partial pressure of CO2 in alveoli // PA_N2 Partial pressure of N2 in alveoli // PA_O2 Partial pressure of O2 in alveoli // PBC_pc Carbaminohemoglobin ION CONC in PULM CAP // PBC_sc Carbaminohemoglobin ION CONC in SYST CAP // PC_CO2 Partial pressure of CO2 in collapsible airways // PC_N2 Partial pressure of N2 in collapsible airways // PC_O2 Partial pressure of O2 in collapsible airways // PCO2_pc Partial pressure of CO2 in pulmonary capillaries // PCO2_sc Partial pressure of CO2 in systemic capillaries // PCO2_isf Partial pressure of CO2 in systemic interstitial fluid // PD_CO2 Partial pressure of CO2 in lung dead space // PD_N2 Partial pressure of N2 in lung dead space // PD_O2 Partial pressure of O2 in lung dead space // pH_pc pH in PULM CAP // pH_sc pH in SYST CAP // pH_isf pH in SYST ISF // PO2_isf Partial pressure of O2 in systemic interstitial fluid // PO2_pc Partial pressure of O2 in pulmonary capillaries // PO2_sc Partial pressure of O2 in systemic capillaries // Paod Distal aorta pressure // Paop Proximal aorta pressure // Pc Collapsible airways pressure // Pcw Chest wall recoil pressure // Pl Lung recoil pressure // Pla Left atrial pressure // Plv Left ventricular pressure // Pmus Applied respiratory muscle pressure // Ppa Pulmonary arterioles pressure // Ppad Distal pulmonary arterial pressure // Ppap Proximal pulmonary arterial pressure // Ppc Pulmonary capillaries pressure // Ppl Pleural pressure // Ppv Pulmonary veins pressure // Pra Right atrial pressure // Prv Right ventricular pressure // Psa Systemic arterial pressure // Psa_a Active systemic arterial pressure component // Psa_p Passive systemic arterial pressure component // Psad Systemic arterioles pressure // Psc Systemic capillaries pressure // Psv Systemic veins pressure // Pvc Vena cava pressure // Pve Viscoelastic lung pressure // Rc Collapsible airways resistance // Rs Small airways resistance // Rsa Resistance of systemic arteries // Ru Upper airways resistance // Rvc Resistance of Vena cava // V_CO2 CO2 production rate in systemic tissue // V_O2 O2 consumption rate in systemic tissue // VA Alveolar volume // Vaod Distal aorta volume // Vaop Proximal aorta volume // Vc Volume of collapsible airways // Vcw Chest wall volume // Vla Volume of left atrium // Vlar Unstressed left atrial volume // Vlv Volume of left ventricle // Vlvr Unstressed left ventricular volume // Vpa Pulmonary arterioles volume // Vpad Distal pulmonary arterial volume // Vpap Proximal pulmonary arterial volume // Vpc Pulmonary capillaries volume // Vpv Pulmonary veins volume // Vra Volume of right atrium // Vrar Unstressed right atrial volume // Vrv Volume of right ventricle // Vrvr Unstressed right ventricular volume // Vsa Systemic arteries volume // Vsad Systemic arterioles volume // Vsc Systemic capillaries volume // Vsv Systemic veins volume // Vtot Total fluid volume in circulation // Vvc Vena cava volume // Vve Lung viscoelastic volume // y Sine wave for ventricle elastance (period = 1/HR2) // ya Sine wave for atrial elastance (period = 1/HR2) //------------------------------------------------------------------------------ // CODE LANGUAGE: MML // CODE: JSim v1.1 import nsrunit; unit conversion on; // Double slashes indicate a comment unit mlSTPD = 1/22400 mole; // math spbhagt{ math CVRESP{ realDomain t sec; t.min=0; t.max=30.0; t.delta=0.01; //----------------------------------------------------------------------------------- // I. PARAMETERS OF V A R Y I N G E L A S T A N C E H E A R T M O D E L //----------------------------------------------------------------------------------- real Vlvrd= 10 ml, // Unstressed end-diastolic left ventricle volume Vlvrs= 10 ml, // Unstressed end-systolic left ventricle volume Vrvrd= 30 ml, // Unstressed end-diastolic right ventricle volume Vrvrs= 30 ml, // Unstressed end-systolic right ventricle volume Vlard= 90 ml, // Unstressed end-diastolic left atrium volume Vlars= 50 ml, // Unstressed end-systolic left atrium volume Vrard= 40 ml, // Unstressed end-diastolic right atrium volume Vrars= 20 ml, // Unstressed end-systolic right atrium volume Rra = 0.001 mmHg*s*ml^-1, // Tricuspid valve resistance Rla = 0.001 mmHg*s*ml^-1, // Mitral valve resistance PRint = 0.15 sec, // Difference in atrial, venticular activation times Emaxlv = 3 mmHg/ml, // Maximum elastance of left ventricle Eminlv = 0.03 mmHg/ml, // Minimum elastance of left ventricle Emaxrv = 1.2 mmHg/ml, // Maximum elastance right ventricle Eminrv = 0.25 mmHg/ml, // Minimum elastance right ventricle Emaxra = 0.2 mmHg/ml, // Maximum elastance right ventricle Eminra = 0.1 mmHg/ml, // Minimum elastance left ventricle Emaxla = 0.75 mmHg/ml, // Maximum elastance right ventricle Eminla = 0.7 mmHg/ml, // Minimum elastance left ventricle y0 = 0, ya0 = 0; // Cut off for sines (elastances) //------------------------------------------------------------------------------- // II. PARAMETERS OF S Y S T E M I C C I R C U L A T I O N //------------------------------------------------------------------------------- // (all parameter values are from Lu et al. 2001) // Resistances: real Rav = 0.005 mmHg*sec*ml^(-1); // Aortic valve resistance // (Lu et al. tricuspid valve value) real Raop = 0.015 mmHg*sec*ml^(-1); // Proximal aortic resistance real Rtaop = 0.04 mmHg*sec*ml^(-1); // Transmural proximal aortic resistance real Rcor = 30.0 mmHg*sec*ml^(-1); // Coronary circulation resistance real Rcrb = 10.0 mmHg*sec*ml^(-1); // Cerebral circulation resistance real Raod = 0.005 mmHg*sec*ml^(-1); // Distal aortic resistance real Rtaod = 0.035 mmHg*sec*ml^(-1); // Transmural distal aortic resistance real Rsad = 0.35 mmHg*sec*ml^(-1); // Systemic arteriolar resistance real Rsc = 0.12 mmHg*sec*ml^(-1); // Systemic capillaries resistance real Rsv = 0.15 mmHg*sec*ml^(-1); // Systemic veins resistance // Compliances: real Caop = 1.2 ml*mmHg^(-1); // Aortic proximal compliance real Caod = 2.3 ml*mmHg^(-1); // Aortic distal compliance real Csad = 3.0 ml*mmHg^(-1); // Systemic arterioles compliance real Csc = 10 ml*mmHg^(-1); // Systemic capillaries compliance // Inertances: real Laop = 0.003 mmHg*sec^2*ml^(-1); // Proximal aorta inertance real Laod = 0.005 mmHg*sec^2*ml^(-1); // Distal aorta inertance // Other parameters: real Kc = 1000 mmHg; // Active vasomotor tone scaling parameter // for systemic arterial pressure real Do = 50 ml; // Active vasomotor tone volume parameter // for systemic arterial pressure real Vsa_o = 210 ml; // Minimal volume of systemic arteries real Vsa_max = 250 ml; // Maximal luminal volume of systemic arteries real Kp1 = 0.03 mmHg; // Passive vasomotor tone scaling parameter // for systemic arterial pressure real Kp2 = 0.20 mmHg*ml^(-2); // Passive vasomotor tone scaling parameter // for systemic arterial pressure real Kr = 0.04 mmHg*sec*ml^(-1); // Pressure scaling constant for systemic // arterial resistance real tau_p = 0.1 ml^(-1); // Passive vasomotor tone constant for systemic // arterial pressure real Kv = 20 mmHg; // Scaling factor for systemic venous pressure real Vmax_sv = 3500 ml; // Maximal volume of lumped systemic veins real D1 = 0.0 mmHg; // Offsetting constant for unstressed and // distended Vena cava pressure real D2 = -5.0 mmHg; // Offsetting constant for partially collapsed // Vena cava pressure real K1 = 0.15 mmHg*ml^(-1); // Scaling constant for unstressed and distended // Vena cava pressure real K2 = 0.4 mmHg; // Scaling constant for partially collapsed Vena // cava pressure real KR = 0.001 mmHg*sec*ml^(-1); // Scaling factor for Vena cava resistance real Ro = 0.025 mmHg*sec*ml^(-1); // Vena cava resistance offset parameter real Vo = 130 ml; // Unstressed volume of Vena cava real Vmax_vc = 350 ml; // Maximum volume of Vena cava real Vmin_vc = 50 ml; // Mimimum volume of Vena cava //--------------------------------------------------------------------------------- // III. PARAMETERS OF P U L M O N A R Y C I R C U L A T I O N //--------------------------------------------------------------------------------- // Resistances: real Rpuv = 0.001 mmHg*sec*ml^(-1); // Pulmonary valve resistance // (Lu et al. tricuspid valve value) real Rtpap = 0.002 mmHg*sec*ml^(-1); // Proximal pulmonary arterial transmural resistance real Rpap = 0.01 mmHg*sec*ml^(-1); // Proximal pulmonary resistance real Rpad = 0.015 mmHg*sec*ml^(-1); // Distal proximal pulmonary resistance real Rps = 1 mmHg*sec*ml^(-1); // Pulmonary shunt resistance real Rpa = 0.03 mmHg*sec*ml^(-1); // Pulmonary arterioles resistance real Rpc = 0.04 mmHg*sec*ml^(-1); // Pulmonary capillaries resistance real Rpv = 0.01 mmHg*sec*ml^(-1); // Pulmonary veins resistance // Compliances: real Ctpap = 1 ml*mmHg^(-1); // Proximal pulmonary arterial compliance real Ctpad = 2 ml*mmHg^(-1); // Distal pulmonary arterial compliance real Cpa = 5 ml*mmHg^(-1); // Pulmonary arterioles compliance real Cpc = 9 ml*mmHg^(-1); // Pulmonary capillaries compliance real Cpv = 21 ml*mmHg^(-1); // Pulmonary veins compliance // Inductors: real Lpa = 0.0001 mmHg*sec^2*ml^(-1);// Pulmonary arterial inertance //--------------------------------------------------------------------------------- // IV. PARAMETERS OF B A R O R E C E P T O R M O D E L //--------------------------------------------------------------------------------- // Baroreceptor Firing Rate is "Nbr." // Firing rate sent to Central Nervous System (CNS). // CNS filters the Nbr signal and outputs efferent firing frequencies: // 1. N_hrv - vagal pathway firing frequency // 2. N_hrs - sympathetic pathway controlling heart rate firing frequency // 3. N_con - sympathetic pathway controlling heart contractility // 4. N_vaso - sympathetic pathway controlling vasomotor tone real a = 0.001 sec; // Time constant for baroreceptor firing rate // NOTE: I made up the value for 'a' without any reference (DB). real a1 = 0.036 sec; // Time constant for baroreceptor firing rate real a2 = 0.0018 sec; // Time constant for baroreceptor firing rate real K = 1.05 sec^(-1)*mmHg^(-1); // Baroreceptor gain (used to account for units) // IVa. HRV pathway real K_hrv = 1.2 dimensionless; // CNS gain for vagal heart rate control real T_hrv = 1.8 sec; // CNS time parameter for vagal heart rate control real L_hrv = 3 sec; // CNS time delay for vagal heart rate control real a_hrv = 0 dimensionless; // Time constant for efferent vagal firing real b_hrv = 1.0 dimensionless; // Time constant for efferent vagal firing real tau_hrv = -0.04 sec; // Time parameter for efferent vagal firing real No_hrv = 110 sec^(-1); // Frequency parameter for efferent vagal firing // IVb. HRS pathway real K_hrs = 1.15 dimensionless; // CNS gain for sympathetic heart rate control real T_hrs = 10.0 sec; // CNS time parameter for sympathetic heart rate control real L_hrs = 3.0 sec; // CNS time delay for sympathetic heart rate control real a_hrs = 0.3 dimensionless; // Time constant for efferent sympathetic heart // rate firing real b_hrs = 0.7 dimensionless; // Time constant for efferent sympathetic heart // rate firing real tau_hrs = 0.09 sec; // Time parameter for efferent sympathetic heart // rate firing real No_hrs = 100 sec^(-1); // Frequency parameter for efferent sympathetic heart // rate firing // IVc. CON pathway real K_con = 1.35 dimensionless;// CNS gain for contractility control real T_con = 10.0 sec; // CNS time parameter for contractility control real L_con = 3.0 sec; // CNS time delay for contractility control real a_con = 0.3 dimensionless; // Time constant for efferent contractility firing real b_con = 0.7 dimensionless; // Time constant for efferent sympathetic // contractility firing real tau_con = 0.04 sec; // Time parameter for efferent sympathetic // contractility firing real No_con = 110 sec^(-1); // Frequency parameter for efferent sympathetic // contractility firing // IVd. VASO pathway real K_vaso = 1.35 dimensionless;// CNS gain for vasomotor tone control real T_vaso = 6.0 sec; // CNS time parameter for vasomotor tone control real L_vaso = 3.0 sec; // CNS time delay for vasomotor tone control real a_vaso = 0.3 dimensionless;// Time constant for efferent vasomotor tone firing real b_vaso = 0.7 dimensionless;// Time constant for efferent vasomotor tone firing real tau_vaso = 0.04 sec; // Time parameter for efferent vasomotor tone // firing real No_vaso = 110 sec^(-1); // Frequency parameter for efferent vasomotor tone // firing // IVe. Heart rate control parameters real h1 = 35 min^(-1); // Heart rate control offset parameter real h2 = 140 min^(-1); // Heart rate control scaling parameter real h3 = 40 min^(-1); // Heart rate control scaling parameter real h4 = 32 min^(-1); // Heart rate control scaling parameter real h5 = 10 min^(-1); // Heart rate control scaling parameter real h6 = 20 min^(-1); // Heart rate control scaling parameter // IVf. Contractility control parameters real amin = -2 dimensionless; // Contractility control offset real bmin = 0.7 dimensionless; // Contractility control offset real Ka = 5 dimensionless; // Contractility control scaling factor real Kb = 0.5 dimensionless; // Contractility control scaling factor //----------------------------------------------------------------------------------- // V. PARAMETERS OF A I R W A Y M E C H A N I C S //----------------------------------------------------------------------------------- real FRC = 3.9 L; // Minimum lung volume real VT = 0.5 L; // Tidal volume real RV = 1.65 L; // Residual volume real TLC = 5.55 L; // Total lung capacity real VD = 0.185 L; // Lung dead space real f = 9.25 1/min; // Breathing frequency (breaths/min) real Ac = 7.09 cmH2O; // Offset for collapsible airways recoil pressure real Acw = 1.4 cmH2O; // Offset for chest wall recoil pressure real Al = 0.2 cmH2O; // Lung recoil pressure scaling parameter real As = 2.2 cmH2O*sec/L; // Small airways resistance scaling parameter real Au = 0.34 cmH2O*sec/L; // Offset for upper airways resistance real Bc = 37.3 cmH2O; // Collapsible airways recoil pressure scaling // parameter real Bcp = 3.73 cmH2O; // Collapsible airways recoil pressure scaling // parameter real Bcw = -3.5 cmH2O; // Chest wall recoil pressure scaling parameter real Bl = -0.5 cmH2O; // Offset for lung recoil pressure real Bs = 0.02 cmH2O*sec/L; // Offset for small airways resistance real Cve = 0.5 L/cmH2O; // Viscoelastic compliance of lung real Kc_air = 0.21 cmH2O*sec/L; // Collapsible airways resistance scaling parameter real Kl = 1.0 1/L; // Lung recoil pressure scaling parameter real Ks = -10.9 dimensionless; // Small airways resistance scaling parameter real Ku = 0.46 cmH2O*sec^2/L^2; // Upper airways resistance scaling parameter real Rve = 1.0 cmH2O*sec/L; // Resistance of lung viscoelastance real Vstar = 5 L; // Small airways resistance parameter real Vcmax = 0.185 L; // Collapsible airwyas resistance parameter real Amus = 2 cmH2O; // Respiratory muscle pressure scaling parameter real tau = 0.1 sec; // Time constant for chest wall volume follower //----------------------------------------------------------------------------------- // VI. PARAMETERS OF G A S E X C H A N G E //----------------------------------------------------------------------------------- //Gas Temperatures and pressures real Tbody = 300 K; //Body temperature real Pstp = 760 mmHg; //STP pressure real Tstp = 273 K; //STP temperature //Saturations real nH = 3.5 dimensionless; //Hill coefficient real P50_O2 = 26.5 mmHg; //P50 of O2 for Hb //Concentrations real CHb = 0.0204 M; // Concentration of O2 or CO2 binding sites on Hb real H = 0.45 dimensionless; // Hematocrit real alphaO2 = 1.36e-6 M/mmHg; // Solubility constant of O2 in plasma // (assume that is is the same for red blood cells) //Interstitial fluid real Visf = 48000 ml; // Value from o2_co2_v1.v11 model by J. Bassingthwaighte real PS = 10 ml/min/g; // Permeability-surface area product for gas exchange real Vcytox = 2.1e-7 mol/min/g; // Vmax for CYTOCHROME OXIDASE in SYST ISF real Kcytox = 1.0e-9 M; // Km for CYTOCHROME OXIDASE in SYST ISF //Saturations real P50_CO2 = 250 mmHg; // P50 of CO2 for Hb //concentrations real alphaCO2 = 24*1.36e-6 M/mmHg; // Solubility constant of CO2 in plasma // (assume that is is the same for red blood cells) real RQ = 0.8 dimensionless; // Respiratory quotient //Partial pressures of gases in humidified atmospheric air real PH2O = 47 mmHg; // Vapor pressure of water at body temperature real Vpcmax = 0.07125 L; // Based on the 4 patients from Liu et.al (1998). //Reference pressure in the Athanasiades model is the airway opening pressure //This pressure is used to make the airway pressures absolute for species conservation extern real Pao(t) mmHg; // External atmospheric pressure extern real r_Pao_O2(t) dimensionless; // Ratio of Pao_O2 to Pao extern real r_Pao_CO2(t) dimensionless; // Ratio of Pao_CO2 to Pao //--------------------------------------------------------------------------------- // XX. PARAMETERS OF B L O O D G A S H A N D L I N G //--------------------------------------------------------------------------------- // MODEL PARAMETERS // Parameters are for a human being of body weight 70 Kg, tissue weight // 60 Kg, cardiac output (F) 6 L/min, Pulmonary capillary volume (V11) = 1.8 L, Alveolar volume // (V12) = 3 L, Systemic capillary volume (V21) = 4.2 L, Systemic interstitial fluid volume // (V22) = 48 L. real kp1 = 0.12 sec^-1, // FORWARD RATE CONST in CO2+H2O REACTION km1 = 89 sec^-1, // BACKWARD RATE CONST in CO2+H2O REACTION K1bgh = 5.5E-4 M, // H2CO3 IONIZATION CONST K2bgh = 1.0e-7 M, // Hb AMINO GROUP IONIZATION CONST K3bgh = 4.0e-7 M, // O2Hb AMINO GROUP IONIZATION CONST K4bgh = 5.525e4 1/M^1.636, // EQUILIBRIUM CONST for O2-Hb BINDinG K5bgh = 2.0e-4, // Hb CARBOMATE IONIZATION CONST K6bgh = 5.0e-5, // O2Hb CARBOMATE IONIZATION CONST kp5 = 5.0e3 M^-1*sec^-1, // FORWARD RATE CONST for Hb CARBOMATE REACTION CFb = 6000, // CARBONIC ANHYDRASE CATALYTIC FACTOR in BLOOD CFt = 5000, // CARBONIC ANHYDRASE CATALYTIC FACTOR in ISF Betab = 0.036 M, // BUFFERING CAPACITY in BLOOD Betat = 0.024 M, // BUFFERING CAPACITY in Interstitial space O2cap = 1.34/22400 mol/g, // O2 CAPACITY of Hb (1.34 ml O2/gm Hb) Hbconc = 150 g/L, // CONC of Hb in BLOOD (0.15 gm Hb/ml BLOOD) Cheme = 0.00923 M, // CONC of hemeHb in BLOOD (4*150/65000 M) HpNorm = 1 M, // H+ NORMALIZATION VARIABLE facid =0.01, // ratio molar acid produced per oxygen mole tweight = 60 kg; real tauHout = 0.1 sec; // Time constant for H+ removal // H+ and HCO3- PS values are from model "o2co2model_v1" aka "o2co2btex" // written by Ranjan Dash (see "Models" section of http://www.physiolme.org) real PS_H = 20 ml/min/g, // PS for H+ across CAP Membrane PS_HCO3 = 20 ml/min/g; // PS for HCO3- across CAP Membrane //----------------------------------------------------------------------------------- // VIII. VARIABLES OF V A R Y I N G E L A S T A N C E H E A R T M O D E L //----------------------------------------------------------------------------------- real y(t), // Sine wave for ventricle elastance (period = 1/HR2) ya(t), // Sine wave for atrial elastance (period = 1/HR2) Elv(t) mmHg/ml, // Time-varying elastance of left ventricle Erv(t) mmHg/ml, // Time-varying elastance of right ventricle Era(t) mmHg/ml, // Time-varying elastance of right atrium Ela(t) mmHg/ml; // Time-varying elastance of left atrium real Vlvr(t) ml, // Unstressed left ventricle volume Vrvr(t) ml, // Unstressed right ventricle volume Vlar(t) ml, // Unstressed left atrial volume Vrar(t) ml; // Unstressed right atrial volume real Fra(t) ml/s, // Flow across tricuspid valve Frv(t) ml/s, // Flow across pulmonary valve Fla(t) ml/s, // Flow across mitral valve Flv(t) ml/s; // Flow across aortic valve real Pra(t) mmHg, // Right atrial transmural pressure Prac(t) mmHg, // Right atrial chamber pressure Prv(t) mmHg, // Right ventricular transmural pressure Prvc(t) mmHg, // Right ventricular chamber pressure Pla(t) mmHg, // Left atrial transmural pressure Plac(t) mmHg, // Left atrial chamber pressure Plv(t) mmHg, // Left ventricular transmural pressure Plvc(t) mmHg; // Left ventricular chamber pressure real Vra(t) ml, // Volume of right atrium Vrv(t) ml, // Volume of right ventricle Vla(t) ml, // Volume of left atrium Vlv(t) ml; // Volume of left ventricle //Initial Conditions when (t=t.min){ Vla = 56.90; Vlv = 101.33; Vra = 57.86; Vrv = 83.26; } extern real Fx(t) ml*sec^(-1); // Blood loss out of left ventricle //--------------------------------------------------------------------------------- // IX. VARIABLES OF S Y S T E M I C C I R C U L A T I O N //--------------------------------------------------------------------------------- // Nonlinear resistances: real Rsa(t) mmHg*sec*ml^(-1); // Resistance of systemic arteries real Rvc(t) mmHg*sec*ml^(-1); // Resistance of Vena cava // Pressures: real Paop1(t) mmHg; // Dummy variable for proximal aorta pressure real Paop2(t) mmHg; // Dummy variable for distal aorta pressure real Paop(t) mmHg; // Proximal aorta pressure real Paod(t) mmHg; // Distal aorta pressure real Psa_a(t) mmHg; // Active systemic arterial pressure component real Psa_p(t) mmHg; // Passive systemic arterial pressure component real Psa(t) mmHg; // Systemic arterial pressure real Psad(t) mmHg; // Systemic arterioles pressure real Psc(t) mmHg; // Systemic capillaries pressure real Psv(t) mmHg; // Systemic veins pressure real Pvc(t) mmHg; // Vena cava transmural pressure real Pvcc(t) mmHg; // Vena cava chamber pressure // Flows: real Faop(t) ml*sec^(-1); // Proximal aorta flow real Faod(t) ml*sec^(-1); // Distal aorta flow real Ftaop(t) ml*sec^(-1); // Proximal aorta radial flow real Ftaod(t) ml*sec^(-1); // Distal aorta radial flow real Fsa(t) ml*sec^(-1); // Systemic arteries flow real Ftsa(t) ml*sec^(-1); // Systemic arteries radial low real Fsad(t) ml*sec^(-1); // Systemic arterioles flow real Ftsad(t) ml*sec^(-1); // Systemic arterioles radial flow real Fsc(t) ml*sec^(-1); // Systemic capillaries flow real Ftsc(t) ml*sec^(-1); // Systemic capillaries radial flow real Fsv(t) ml*sec^(-1); // Systemic veins flow real Ftsv(t) ml*sec^(-1); // Systemic veins radial flow real Fvc(t) ml*sec^(-1); // Vena cava flow real Ftvc(t) ml*sec^(-1); // Vena cava radial flow real Fcor(t) ml*sec^(-1); // Coronary flow real Fcrb(t) ml*sec^(-1); // Cerebral flow // Volumes: real Vaop(t) ml; // Proximal aorta volume real Vaod(t) ml; // Distal aorta volume real Vsa(t) ml; // Systemic arteries volume real Vsad(t) ml; // Systemic arterioles volume real Vsc(t) ml; // Systemic capillaries volume real Vsv(t) ml; // Systemic veins volume real Vvc(t) ml; // Vena cava volume real Vtot(t) ml; // Total fluid volume in circulation // Initial values: when(t=t.min) { Faop = 17.83; Faod = 73.30; Vaop = 107.20; Vaod = 206.42; Vsa = 226.07; Vsad = 179.10; Vsc = 310.21; Vsv = 3254.17; Vvc = 167.55; } //--------------------------------------------------------------------------------- // X. VARIABLES OF P U L M O N A R Y C I R C U L A T I O N //--------------------------------------------------------------------------------- //Pressures: real Ppap(t) mmHg; // Proximal pulmonary arterial pressure real Ppap1(t) mmHg; // Dummy variable for Ppap real Ppap2(t) mmHg; // Dummy variable for Ppap real Ppad(t) mmHg; // Distal pulmonary arterial pressure real Ppa(t) mmHg; // Pulmonary arterioles pressure real Ppc(t) mmHg; // Pulmonary capillaries pressure real Ppv(t) mmHg; // Pulmonary veins pressure real Vpap(t) ml; // Proximal pulmonary arterial volume real Vpad(t) ml; // Distal pulmonary arterial volume real Vpa(t) ml; // Pulmonary arterioles volume real Vpc(t) ml; // Pulmonary capillaries volume real Vpv(t) ml; // Pulmonary veins volume real Fpap(t) ml*sec^(-1); // Proximal pulmonary arterial flow real Ftpap(t) ml*sec^(-1); // Proximal pulmonary arterial radial flow real Ftpad(t) ml*sec^(-1); // Distal pulmonary arterial radial flow real Fpad(t) ml*sec^(-1); // Distal pulmonary arterial flow real Fps(t) ml*sec^(-1); // Pulmonary shunt flow real Fpa(t) ml*sec^(-1); // Pulmonary arterioles flow real Ftpa(t) ml*sec^(-1); // Pulmonary arterioles radial flow real Fpc(t) ml*sec^(-1); // Pulmonary capillaries flow real Ftpc(t) ml*sec^(-1); // Pulmonary capillaries radial flow real Fpv(t) ml*sec^(-1); // Pulmonary veins flow real Ftpv(t) ml*sec^(-1); // Pulmonary veins radial flow // Initial values: when(t=t.min) { Fpap = 8.26; Vpap = 13.65; Vpad = 27.10; Vpa = 65.92; Vpc = 103.54; Vpv = 156.73; } //--------------------------------------------------------------------------------- // XI. VARIABLES OF B A R O R E C E P T O R M O D E L //--------------------------------------------------------------------------------- real Nbr(t) sec^(-1); // Instantaneous firing frequency of baroreceptor real Nbr_t(t) sec^(-2); // Time derivative of baroreceptor firing rate // Initial values: when(t=t.min) { Nbr = 91.71; Nbr_t = -25.46; } real N_hrv(t) sec^(-1); // Vagal discharge rate at CNS controlling heart rate real F_hrv(t) dimensionless; // Normalized vagal discharge rate // Initial Values: when(t=t.min) { N_hrv = 114.01; } real N_hrs(t) sec^(-1); // Sympathetic discharge rate at CNS controlling heart rate real F_hrs(t) dimensionless; // Normalized discharge rate for sympathetic heart // rate control // Initial Values: when(t=t.min) { N_hrs = 110.50; } real N_con(t) sec^(-1); // Sympathetic discharge rate at CNS for contractility real F_con(t) dimensionless; // Normalized discharge rate for contractility control // Initial Values: when(t=t.min) { N_con = 129.71; } real N_vaso(t) sec^(-1); // Sympathetic discharge rate at CNS controlling // vasomotor tone real F_vaso(t) dimensionless; // Normalized discharge rate for vasomotor control // Initial Values: when(t=t.min) { N_vaso = 129.88; } real afs_con(t) dimensionless; // Dynamic heart contractility scaling function real bfs_con(t) dimensionless; // Contractility variable to alter Lu et al. (2001) // activation function realState afs_con2(t) dimensionless; // Discrete heart contractility scaling function // Initial Values: when(t=t.min) { afs_con2=afs_con; } real HR(t) min^(-1); // Dynamic heart rate realState HR2(t) min^(-1); // Discrete heart rate function // Initial Values: when(t=t.min) { HR2 = HR; } // Variables used for discrete heart rate function realState tshift(t) sec; // Time offset for heart rate real phi(t) dimensionless; // Offset for heart rate //Initial Values: when(t=t.min) { tshift = t.min; } //--------------------------------------------------------------------------------- // XII. VARIABLES OF A I R W A Y M E C H A N I C S //--------------------------------------------------------------------------------- real Rs(t) cmH2O*sec/L; // Small airways resistance real Rc(t) cmH2O*sec/L; // Collapsible airways resistance real Ru(t) cmH2O*sec/L; // Upper airways resistance real Vc(t) L; // Volume of collapsible airways real VA(t) L; // Alveolar volume real Vve(t) L; // Lung viscoelastic volume real Vcw(t) L; // Chest wall volume real dV(t) L; // Chest wall volume follower to get flow real Pcw(t) cmH2O; // Chest wall recoil pressure real Pl(t) cmH2O; // Lung recoil pressure real PA(t) cmH2O; // Alveolar pressure real Pc(t) mmHg; // Collapsible airways pressure real Pve(t) cmH2O; // Viscoelastic lung pressure real Ppl(t) mmHg; // Pleural pressure real Pmus(t) cmH2O; // Applied respiratory muscle pressure // I HAVE BEEN UNABLE TO LOCATE A NICE SET OF // INITIAL CONDITIONS. I'VE MADE A GUESS ABOUT // THESE PARAMETERS, BUT KNOW THAT THEY ARE FAR // OFF FROM WHAT SHOULD BE. -MK when (t=t.min) { dV = 3.03; VA = 2.81; Vve = 0.16; Vc = 0.07; } //--------------------------------------------------------------------------------- // XIII. VARIABLES OF G A S E X C H A N G E //--------------------------------------------------------------------------------- real PO2_isf(t) mmHg; real dSHbO2dPO2_pc(t) mmHg^-1; real dSHbO2dPO2_sc(t) mmHg^-1; real SHbO2_sc(t) dimensionless; // Oxygen saturation of Hb in sys cap real SHbO2_pc(t) dimensionless; // Oxygen saturation of Hb in Pulm cap real CtO2_pc(t) M; real CtO2_sc(t) M; real CtO2_isf(t) M; // Needed for GO2isf - added by mneal //Partial pressures real PO2_pc(t) mmHg; real PO2_sc(t) mmHg; real DL_O2(t) mlSTPD/s/mmHg; real V_O2(t) mole/min; real O2flux(t) ml/min; real PCO2_pc(t) mmHg; real PCO2_sc(t) mmHg; real PCO2_isf(t) mmHg; when(t=t.min) { PO2_pc = 87.70; PO2_sc = 34.38; PO2_isf = 19.00; } when(t=t.min) { PCO2_pc = 42.97; PCO2_sc = 61.02; PCO2_isf = 61.52; } real SHbCO2_pc(t) dimensionless; //CO2 saturation of Hb in Pulm cap real SHbCO2_sc(t) dimensionless; //CO2 saturation of Hb in sys cap real dSHbCO2dPCO2_pc(t) mmHg^-1; real dSHbCO2dPCO2_sc(t) mmHg^-1; real CtCO2_pc(t) M; real CtCO2_sc(t) M; real CtCO2_isf(t) M; // Needed for B_isf - added by mneal real DL_CO2(t) mlSTPD/s/mmHg; real V_CO2(t) mole/min; real CO2flux(t) ml/min; //Flows through the airways real QdotCA(t) ml/s; // Flow through Rs real QdotDC(t) ml/s; // Flow through Rc real QdotED(t) ml/s; // Flow through Ru real Pao_O2(t) mmHg; real Pao_CO2(t) mmHg; real Pao_N2(t) mmHg; //Partial pressures at different nodes: real PD_O2(t) mmHg; real PC_O2(t) mmHg; real PA_O2(t) mmHg; real PD(t) cmH2O; //total rigid Dead space pressure when (t=t.min) { PD_O2 = 102.92; PC_O2 = 99.02; PA_O2 = 98.69; } //Carbon dioxide //Partial pressures at different nodes: real PD_CO2(t) mmHg; // Dead space real PC_CO2(t) mmHg; // Collapsible airways real PA_CO2(t) mmHg; // Alveoli //Species conservation ODE's //CO2 when (t=t.min) { PD_CO2 = 39.24; PC_CO2 = 42.49; PA_CO2 = 42.75; } //Nitrogen real PD_N2(t) mmHg; real PC_N2(t) mmHg; real PA_N2(t) mmHg; //--------------------------------------------------------------------------------- // XX. VARIABLES OF B L O O D G A S H A N D L I N G //--------------------------------------------------------------------------------- real DVRO11(t), // COEFF of dpO11/dt in d(pO11*VRO11)/dt DVRO21(t), // COEFF of dpO21/dt in d(pO21*VRO21)/dt GO2isf(t) ml/min/g, // GULOSITY FOR O2 CONSUMPTION in TISSUE PBC_pc(t) M, // HbNHCOO- ION CONC in PULM CAP PBC_sc(t) M, // HbNHCOO- ION CONC in SYST CAP B_pc(t) M, // HCO3- ION CONC in PULM CAP B_sc(t) M, // HCO3- ION CONC in SYST CAP B_isf(t) M, // HCO3- ION CONC in SYST ISF H_pc(t) M, // H+ ION CONC in PULM CAP H_sc(t) M, // H+ ION CONC in SYST CAP H_isf(t) M, // H+ ION CONC in SYST ISF pH_pc(t), // pH in PULM CAP pH_sc(t), // pH in SYST CAP pH_isf(t); // pH in SYST ISF real Hout(t) M; // Variable for H+ ion sink (crude kidney) extern real Hin(t) M/sec; // Infusion of acid into systemic circulation // INITIAL CONDITIONS when(t=t.min){ PBC_pc = 6.12e-3; // HBNHCOO- in PULM CAP PBC_sc = 6.59e-3; // HBNHCOO- in SYST CAP B_pc = 0.03; // HCO3- in PULM CAP B_sc = 0.04; // HCO3- in SYST CAP B_isf = 0.02; // HCO3- in SYST ISF H_pc = 4.16e-8; // H+ in PULM CAP H_sc = 7.4e-8; // H+ in SYST CAP H_isf = 7.3e-8; // H+ in SYST ISF Hout = 1.9E-5; // H+ lost to crude kidney } //--------------------------------------------------------------------------------- // XV. EQUATIONS OF V A R Y I N G E L A S T A N C E H E A R T M O D E L //--------------------------------------------------------------------------------- //--------------------------------------------------------------------------------- // REFERENCE EQUATIONS: // Eq. A) Flow (mL/unit time) = change in volume / change in time // Basis: Definition of flow // Eq. B) Compliance = Change in volume / Change in pressure // Basis: Fluid analog of capacitance // Eq. C) Pressure drop = Resistance * Flow // Basis: Fluid analog of Ohm's Law // Eq. D) (Sum of flows entering junction = sum of flows leaving junction) // Basis: Kirchhoff Junction rule // Eq. E) Pressure drop = (change in Flow/change in time)*Inertance // Basis: Fluid analog of inductance // Eq. F) Elastance = Compliance^(-1) // Basis: Definition of elastance //--------------------------------------------------------------------------------- // The driving function for time varying ventric. elastances is a // rectified sine wave, y(t), cut off at y0, biased to Emin and // scaled to range between Emin and Emax. Similarly, sine for atria is ya(t) event (t+t.delta>=(tshift+(1/HR2))) { // Heart rate and contractility update discretely HR2 = HR; tshift = t; afs_con2=afs_con; } phi = 2*PI*tshift*HR2; // Right atrial elastance rise precedes right ventricle by PRint y = sin((2*PI*HR2*t) - phi); ya = sin((2*PI*HR2*(t+PRint)) - phi); Era = if (ya>ya0) (Emaxra-Eminra)*(ya-ya0)/(1-ya0)+Eminra else Eminra; Vrar = ((Emaxra-Era)/Emaxra)*(Vrard-Vrars) + Vrars; // Right ventricular elastance has same timing as for LV Erv = if (y>y0) (Emaxrv-Eminrv)*(y-y0)/(1-y0) + Eminrv else Eminrv; Vrvr = ((Emaxrv-Erv)/Emaxrv)*(Vrvrd-Vrvrs) + Vrvrs; //Left atrial elastance rise precedes left ventricle by PRint Ela = if (ya>ya0) (Emaxla-Eminla)*(ya-ya0)/(1-ya0) + Eminla else Eminla; Vlar = ((Emaxla-Ela)/Emaxla)*(Vlard-Vlars) + Vlars; // Left ventricle elastance defines timing for heart Elv = if (y>y0) (Emaxlv-Eminlv)*(y-y0)/(1-y0) + Eminlv else Eminlv; Vlvr = ((Emaxlv-Elv)/Emaxlv)*(Vlvrd-Vlvrs) + Vlvrs; // Calculate pressures, volumes and flows in each segment //------------------------------------------------------- // Left Atrium, la, and its Contraction: Vla:t = Fpv - Fla; // Eq. A,D Fla = if(Plac > Plvc) (Plac-Plvc)/(Rla) // Eq. C else 0; // Mitral valve closed Pla = (Vla-Vlar)*Ela; // Eq. B,F Plac = Pla + Ppl; // Left Ventricle, lv, and its contraction Vlv:t = Fla-Flv-Fx; // Eq. A,D Blood loss function here Flv = if(Plvc > Paop) (Plvc-Paop)/Rav // Eq. C else 0; // Aortic valve closed // Left ventricle pressure Plv = if (y>y0) afs_con2*(Vlv-Vlvr)*Elv // Contractility effects systolic elastance else (Vlv-Vlvr)*Elv; // Eq. B,F Plvc = Plv + Ppl; // Right atrium, ra, and its Contraction: Vra:t = Fvc - Fra + Fcor; // Eq. A,D Fra = if(Prac > Prvc) (Prac-Prvc)/Rra // Eq. C else 0; // Tricuspid valve closed Pra = (Vra-Vrar)*Era; // Eq. B,F Prac = Pra + Ppl; // Right Ventricle, rv and its contraction: Vrv:t = Fra - Frv; // Frv is pulm valve flow Frv = if (Prvc > Ppap) (Prvc-Ppap)/Rpuv // Eq. C else 0; // Pulm Valve closed Prv = if (y>y0) afs_con2*(Vrv-Vrvr)*Erv // Contractility effects systolic elastance else (Vrv-Vrvr)*Erv; // Eq. B,F Prvc = Prv + Ppl; //--------------------------------------------------------------------------------- // XVI. EQUATIONS OF S Y S T E M I C C I R C U L A T I O N //--------------------------------------------------------------------------------- // Nonlinear resistances: Rsa = (Kr*exp(4*F_vaso)) + (Kr*(Vsa_max/Vsa)^2);// Sys. arteries: Lu et al. Eq.(16) Rvc = (KR*(Vmax_vc/Vvc)^2) + Ro; // Vena cava: Lu et al. Eq.(4) Paop1 = (-Rtaop*Caop*Rcor*Rav*Rcrb*Faop+Vaop*Rcor*Rav*Rcrb+ Rtaop*Rav*Rcrb*Caop*Pvc+Rtaop*Rav*Rcor*Caop*Pra+Rtaop*Rcor*Rcrb*Caop*Plv)/ (Caop*(Rcor*Rav*Rcrb+Rav*Rcrb*Rtaop+Rcor*Rcrb*Rtaop+Rav*Rcor*Rtaop));// Eqs. B,C,D Paop2 = (Rcor*Rcrb*Vaop-Rtaop*Caop*Rcor*Rcrb*Faop+ Rtaop*Caop*Rcrb*Pra+Rtaop*Caop*Rcor*Pvc)/ (Rcor*Rcrb+Rcrb*Rtaop+Rtaop*Rcor)/Caop; Paop = if (Plv>Paop1) Paop1 else Paop2; // Input to arterial model // Pressures: Paod = Ftaod*Rtaod + Vaod/Caod; // Eq. B,C Psa_a = Kc*log( ((Vsa-Vsa_o)/Do) + 1 ); // Lu et al. Eq. (13) Psa_p = Kp1*exp(tau_p*(Vsa-Vsa_o)) + Kp2*(Vsa-Vsa_o)^2; // Lu et al. Eq. (14) Psa = F_vaso*Psa_a + (1-F_vaso)*Psa_p; // Lu et al. Eq. (15) Psad = Vsad/Csad; // Eq. B Psc = Vsc/Csc; // Eq. B Psv = -Kv*log((Vmax_sv/Vsv)-0.99); // Lu et al. Eq. (1) Pvcc = if (Vvc>Vo) D2+K2*exp(Vo/Vmin_vc)+K1*(Vvc-Vo) // Lu et al. Eq. (2) else D2+K2*exp(Vvc/Vmin_vc); // Lu et al. Eq. (3) Pvc = Pvcc - Ppl; // Flows: Fcor = (Paop-Pra)/Rcor; // Eq. C Fcrb = (Paop-Pvc)/Rcrb; // Eq. C Fsa = (Psa-Psad)/Rsa; // Eq. C Fsad = (Psad-Psc)/Rsad; // Eq. C Fsc = (Psc-Psv)/Rsc; // Eq. C Fsv = (Psv-Pvc)/Rsv; // Eq. C Fvc = (Pvc-Pra)/Rvc; // Eq. C // Transmural flows: Ftaop = Flv - Fcrb - Fcor - Faop; // Eq. D Ftaod = Faop - Faod; // Eq. D Ftsa = Faod - Fsa; // Eq. D Ftsad = Fsa - Fsad; // Eq. D Ftsc = Fsad - Fsc; // Eq. D Ftsv = Fsc - Fsv; // Eq. D Ftvc = Fsv + Fcrb - Fvc; // Eq. D // Differential equations: Faop:t = (Paop - Faop*Raop - Paod) / Laop; // Eq. E Faod:t = (Paod - Faod*Raod - Psa) / Laod; // Eq. E Vaop:t = Ftaop; // Eq. A Vaod:t = Ftaod; // Eq. A Vsa:t = Ftsa; // Eq. A Vsad:t = Ftsad; // Eq. A Vsc:t = Ftsc; // Eq. A Vsv:t = Ftsv; // Eq. A Vvc:t = Ftvc; // Eq. A Vtot = Vra+Vrv+Vpap+Vpad+Vpa+Vpc+Vpv+Vla+Vlv+Vaop+Vaod +Vsa+Vsad+Vsc+Vsv+Vvc; // Total fluid volume // in circulation //--------------------------------------------------------------------------------- // XVII. EQUATIONS OF P U L M O N A R Y C I R C U L A T I O N //--------------------------------------------------------------------------------- Ppap1 = (Rpuv*Vpap + Ctpap*Prv*Rtpap - Rpuv*Rtpap*Ctpap*Fpap) // Eqs. B,C,D / (Ctpap*(Rtpap+Rpuv)); Ppap2 = (Vpap - Ctpap*Rtpap*Fpap)/Ctpap; // Eqs. B,C,D Ppap = if (Prv>Ppap1) Ppap1 else Ppap2; Ppad = Vpad/Ctpad; // Eq. B Ppa = Vpa/Cpa; // Eq. B Ppc = Vpc/Cpc; // Eq. B Ppv = Vpv/Cpv; // Eq. B Fpad = (Ppad-Ppa)/Rpad; // Eq. C Fps = (Ppa-Ppv)/Rps; // Eq. C Fpa = (Ppa-Ppc)/Rpa; // Eq. C Fpc = (Ppc-Ppv)/Rpc; // Eq. C Fpv = (Ppv-Pla)/Rpv; // Eq. C Ftpad = Fpap - Fpad; // Eq. D Ftpap = Frv - Fpap; // Eq. D Ftpa = Fpad - Fps - Fpa; // Eq. D Ftpc = Fpa - Fpc; // Eq. D Ftpv = Fpc + Fps - Fpv; // Eq. D Vpap:t = Ftpap; // Eq. A Vpad:t = Ftpad; // Eq. A Vpa:t = Ftpa; // Eq. A Vpc:t = Ftpc; // Eq. A Vpv:t = Ftpv; // Eq. A Fpap:t = (Ppap - Ppad - Fpap*Rpap)/Lpa; // Eq. E //--------------------------------------------------------------------------------- // XVIII. EQUATIONS OF B A R O R E C E P T O R M O D E L //--------------------------------------------------------------------------------- Nbr:t = Nbr_t; // The governing equation for Nbr is adopted // from Spickler et al. (Kezdi and Geller, 1967). // Governing equation: (a2*a*(Nbr_t:t)) + ((a2+a)*Nbr_t) + Nbr = (K*Paop) + (a1*K*(Paop:t)); // Spickler et al. p. 34, also Lu et al. Eq. (6) // The general form of the equation determining // efferent responses to the baroreceptor: // T_x*(N_x:t) + N_x = K_x*Nbr(t-L_x) // Lu et al. Eq. in CNS box of Fig 2 // and // F_x = a_x + b_x / (exp(tau_x*(N_x-No_x)) + 1.0), // Lu et al. Eq. (7) // where the subscript "x" denotes the pathway name. // Discharge frequency controlling heart rate (vagal pathway) N_hrv:t = if (t>L_hrv) (-N_hrv + (K_hrv*Nbr(t-L_hrv)))/T_hrv else 0; F_hrv = a_hrv + (b_hrv / (exp(tau_hrv*(N_hrv-No_hrv)) + 1.0)); // Discharge frequency controlling heart rate (sympathetic) N_hrs:t = if (t>L_hrs) (-N_hrs + (K_hrs*Nbr(t-L_hrs)))/T_hrs else 0; F_hrs = a_hrs + (b_hrs / (exp(tau_hrs*(N_hrs-No_hrs)) + 1.0)); // Discharge frequency controlling contractility of heart N_con:t = if (t>L_con) (-N_con + (K_con*Nbr(t-L_con)))/T_con else 0; F_con = a_con + (b_con / (exp(tau_con*(N_con-No_con)) + 1.0)); // Discharge frequency controlling vasomotor tone N_vaso:t = if (t>L_vaso) (-N_vaso + (K_vaso*Nbr(t-L_vaso)))/T_vaso else 0; F_vaso = a_vaso + (b_vaso / (exp(tau_vaso*(N_vaso-No_vaso)) + 1.0)); // Heart rate control HR = (h1 + (h2*F_hrs)-(h3*F_hrs^2)-(h4*F_hrv)+ (h5*F_hrv^2)-(h6*F_hrv*F_hrs)); // Lu et al. Eq. (8) // Contractility control afs_con = amin + (Ka*F_con); bfs_con = bmin + (Kb*F_con); //--------------------------------------------------------------------------------- // XIX. EQUATIONS OF A I R W A Y M E C H A N I C S //--------------------------------------------------------------------------------- // AIRWAY RESISTANCES AND COMPLIANCES COME FROM TABLE 2 // IN THE MANUSCRIPT. THERE ARE SEVERAL MINOR CORRECTIONS // NOTED BELOW. // // AIRWAYS RESISTANCES Rc = Kc_air*(Vcmax/Vc)^2; Rs = As * exp(Ks * (VA - RV)/(Vstar - RV)) + Bs; Ru = Au + Ku*abs(dV:t); // COMPLIANT RECOIL PRESSURES // SIGNS FOR Acw & Bcw ARE INVERTED FROM MANUSCRIPT Pcw = -Acw + Bcw*log((TLC-RV)/(Vcw-RV) - 0.999); Pl = Al * exp(Kl * VA) + Bl; Pmus = Amus*sin(2*PI*f*t) + Amus; // SIGN FOR Bc INVERTED FROM MANUSCRIPT // AND A LAST ELSE CONDITION (Vc/Vcmax < Vccrit) IGNORED Pc = if (Vc/Vcmax<0.5) Ac-Bc*(Vc/Vcmax - 0.7)^2 else 5.6-Bcp*log(Vcmax/Vc - 0.999); Pve = Vve/Cve; // UPPER AIRWAYS RESISTANCE DEPENDENT ON FLOW // I'VE PUT IN A Vcw FOLLOWER dV, THOUGH THIS // IS PROBABLY NOT NECESSARY IN THIS MODEL ANY // LONGER-- WE DO NOT HAVE A Vcw:t TERM // ON THE LHS OF ANY EQUATION. dV:t = (Vcw-dV)/tau; Vcw = VA + VD; // PRESSURE, RESISTANCE, FLOW RELATIONS // THE FOLLOWING EQUATION ARE DETERMINED BY SOLVING // THE MATRIX (EQN 7) FOR FLOWS IN STATE FORM. Vc:t = (Pmus-Pc-Pcw)/(Ru+Rs) - (Pc-Pl-Pve)/Rs; VA:t = (Pc-Pl-Pve)/Rs - (Pstp*Tbody/((PA + (760 mmHg))*Tstp))*(O2flux+CO2flux); // accounts for // gas exchange influence // on alveolar volume Vve:t = -Pve/Rve - (Pc-Pl-Pve)/Rs; // SOME SIMPLE CALCULATIONS TO FIND PRESSURES // WE WOULD NORMALLY LIKE TO REFERENCE FROM THE MANUSCRIPT Ppl = Pcw - Pmus; PA = Ppl + Pl + Pve; PD = Pc+Ppl + Rc*(Vc:t + (Pc+Ppl-PA)/Rs); //--------------------------------------------------------------------------------- // XX. EQUATIONS OF G A S E X C H A N G E //--------------------------------------------------------------------------------- //--------------------------OXYGEN------------------------------------------------- SHbO2_pc = (PO2_pc/P50_O2)^nH/(((PO2_pc/P50_O2)^nH) + 1); // Saturation in pc SHbO2_sc = (PO2_sc/P50_O2)^nH/(((PO2_sc/P50_O2)^nH) + 1); // Saturation in sc dSHbO2dPO2_pc = nH*(PO2_pc^-1)*(PO2_pc/P50_O2)^nH/(((PO2_pc/P50_O2)^nH) + 1)^2; dSHbO2dPO2_sc = nH*(PO2_sc^-1)*(PO2_sc/P50_O2)^nH/(((PO2_sc/P50_O2)^nH) + 1)^2; CtO2_pc = alphaO2*PO2_pc + CHb*SHbO2_pc*H; // Concentration of O2 in pulmonary // capillaries (Hb-bound and in plasma) CtO2_sc = alphaO2*PO2_sc + CHb*SHbO2_sc*H; CtO2_isf = alphaO2*PO2_isf; DL_O2 = ((sqrt(Vpc/Vpcmax))*((0.397 mlSTPD/s/mmHg)+((0.0085 mlSTPD/s/mmHg^2)*PO2_pc)- ((0.00013 mlSTPD/s/mmHg^3)*(PO2_pc^2)))+((5.1e-7 mlSTPD/s/mmHg^4)*(PO2_pc^3))); V_O2 = if (PO2_isf<=0) 0 else (Vcytox*tweight*alphaO2*PO2_isf)/(Kcytox + (alphaO2*PO2_isf)); // O2 consumption //Oxygen Partial Pressures in systemic and pulm caps and ISF ODEs //Oxygen STP volume diffusion flux into pulmonary capillary O2flux = DL_O2*(PA_O2-PO2_pc)*(22400 ml/mole); PO2_pc:t = (Fpc*(CtO2_sc-CtO2_pc) + DL_O2*(PA_O2-PO2_pc))/ (Vpc*(alphaO2+H*CHb*dSHbO2dPO2_pc)); PO2_sc:t = (Fsc*(CtO2_pc-CtO2_sc) + PS*tweight*alphaO2*(PO2_isf-PO2_sc))/ (Vsc*(alphaO2+H*CHb*dSHbO2dPO2_sc)); PO2_isf:t = PS*tweight*(PO2_sc-PO2_isf)/Visf - V_O2/(alphaO2*Visf); //--------------------------CARBON DIOXIDE---------------------------------------- SHbCO2_pc = (PCO2_pc/P50_CO2)/((PO2_pc/P50_O2) + 1); SHbCO2_sc = (PCO2_sc/P50_CO2)/((PCO2_sc/P50_CO2) + 1); dSHbCO2dPCO2_pc = (PCO2_pc^-1)*(PCO2_pc/P50_CO2)/((PCO2_pc/P50_CO2)+ 1)^2; dSHbCO2dPCO2_sc = (PCO2_sc^-1)*(PCO2_sc/P50_CO2)/((PCO2_sc/P50_CO2)+ 1)^2; CtCO2_pc = alphaCO2*PCO2_pc + CHb*SHbCO2_pc*H; CtCO2_sc = alphaCO2*PCO2_sc + CHb*SHbCO2_sc*H; CtCO2_isf = alphaCO2*PCO2_isf; DL_CO2 = (sqrt(Vpc/Vpcmax))*(16.67 mlSTPD/s/mmHg); V_CO2 = RQ*V_O2; //CO2 Partial Pressures in systemic and pulm caps and ISF ODEs //CO2 STP volume diffusion flux into pulmonary capillary CO2flux = DL_CO2*(PA_CO2-PCO2_pc)*(22400 ml/mole); PCO2_pc:t = (Fpc*(CtCO2_sc-CtCO2_pc) + DL_CO2*(PA_CO2-PCO2_pc))/ (Vpc*(alphaCO2+H*CHb*dSHbCO2dPCO2_pc)); PCO2_sc:t = (Fsc*(CtCO2_pc-CtCO2_sc) + PS*tweight*alphaCO2*(PCO2_isf-PCO2_sc))/ (Vsc*(alphaCO2+H*CHb*dSHbCO2dPCO2_sc)); PCO2_isf:t = PS*tweight*(PCO2_sc-PCO2_isf)/Visf + V_CO2/(alphaCO2*Visf); //Flows through the airways QdotCA = (Pc+Ppl-PA)/Rs; QdotDC = (PD-(Pc+Ppl))/Rc; QdotED = QdotDC; Pao_O2 = (Pao - PH2O)*r_Pao_O2; Pao_CO2 = (Pao-PH2O)*r_Pao_CO2; Pao_N2 = (Pao-PH2O) - Pao_O2 - Pao_CO2; //Species conservation ODEs //--------------------------------------------------------------------------------- //Oxygen PD_O2:t = if (VA:t > 0) (QdotED*Pao_O2 - QdotDC*PD_O2)/VD else (QdotED*PD_O2 - QdotDC*PC_O2)/VD; PC_O2:t = if (VA:t > 0) (QdotDC*PD_O2 - QdotCA*PC_O2 - PC_O2*Vc:t)/Vc else (QdotDC*PC_O2 - QdotCA*PA_O2 - PC_O2*Vc:t)/Vc; PA_O2:t = if (VA:t > 0) (QdotCA*PC_O2 - PA_O2*VA:t)/VA -(Pstp*Tbody/Tstp)*(O2flux)/VA else (QdotCA*PA_O2 - PA_O2*VA:t)/VA -(Pstp*Tbody/Tstp)*(O2flux)/VA; //CO2 PD_CO2:t = if (VA:t > 0) (QdotED*Pao_CO2 - QdotDC*PD_CO2)/VD else (QdotED*PD_CO2 - QdotDC*PC_CO2)/VD; PC_CO2:t = if (VA:t > 0) (QdotDC*PD_CO2 - QdotCA*PC_CO2 - PC_CO2*Vc:t)/Vc else (QdotDC*PC_CO2 - QdotCA*PA_CO2 - PC_CO2*Vc:t)/Vc; PA_CO2:t = if (VA:t > 0) (QdotCA*PC_CO2 - PA_CO2*VA:t)/VA -(Pstp*Tbody/Tstp)*(CO2flux)/VA else (QdotCA*PA_CO2 - PA_CO2*VA:t)/VA -(Pstp*Tbody/Tstp)*(CO2flux)/VA; //Nitrogen PD_N2 = (PD + Pao) - PD_O2 - PD_CO2; PC_N2 = (Pc+Ppl+Pao) - PC_O2 - PC_CO2; PA_N2 = (PA + Pao) - PA_O2 - PA_CO2; //--------------------------------------------------------------------------------- // XX. EQUATIONS OF B L O O D G A S H A N D L I N G //--------------------------------------------------------------------------------- GO2isf = if (((Vcytox)/(Kcytox + CtO2_isf))>(51ml/min/g)) (51ml/min/g) else (Vcytox)/(Kcytox + CtO2_isf); // O2 gulosity in isf ( =tissue) // - cap set by mneal to prevent // model from blowing during stress PBC_pc:t = (Fpc/Vpc)*(PBC_sc-PBC_pc) + kp5*CtCO2_pc*(Cheme-PBC_pc)*(SHbO2_pc/(1+H_pc/K3bgh) + (1-SHbO2_pc)/(1+H_pc/K2bgh)) - kp5*PBC_pc*H_pc*(SHbO2_pc/K6bgh + (1-SHbO2_pc)/K5bgh); PBC_sc:t = (Fsc/Vsc)*(PBC_pc-PBC_sc) + kp5*CtCO2_sc*(Cheme-PBC_sc)*(SHbO2_sc/(1+H_sc/K3bgh) + (1-SHbO2_sc)/(1+H_sc/K2bgh)) - kp5*PBC_sc*H_sc*(SHbO2_sc/K6bgh + (1-SHbO2_sc)/K5bgh); B_pc:t = (Fpc/Vpc)*(B_sc-B_pc) + CFb*(kp1*CtCO2_pc - km1*B_pc*H_pc/K1bgh); B_sc:t = (Fsc/Vsc)*(B_pc-B_sc) + CFb*(kp1*CtCO2_sc - km1*B_sc*H_sc/K1bgh); B_isf:t = (PS_HCO3*tweight/Visf)*(B_pc-B_isf) + CFt*(kp1*CtCO2_isf - km1*B_isf*H_isf/K1bgh); DVRO11 = 1 + dSHbO2dPO2_pc*O2cap*Hbconc/alphaO2; DVRO21 = 1 + dSHbO2dPO2_sc*O2cap*Hbconc/alphaO2; Hout:t = if (H_pc > 4E-8) (H_pc - 4E-8)/tauHout else 0; H_pc:t = (Fpc/Vpc)*(H_sc-H_pc) + (2.303/Betab)*H_pc*(CFb*(kp1*CtCO2_pc - km1*H_pc*B_pc/K1bgh) + 1.5*(PBC_pc:t) - 0.6*alphaO2*DVRO11*(PO2_pc:t)) -Hout:t; H_sc:t = (Fsc/Vsc)*(H_pc-H_sc) + (2.303/Betab)*H_sc*(CFb*(kp1*CtCO2_sc - km1*H_sc*B_sc/K1bgh) + 1.5*(PBC_sc:t) - 0.6*alphaO2*DVRO21*(PO2_sc:t)) + Hin; H_isf:t = (PS_H*tweight/Visf)*(H_pc-H_isf) + (2.303/Betat)*H_isf*CFt*(kp1*CtCO2_isf - km1*H_isf*B_isf/K1bgh) + (facid*GO2isf*tweight*CtO2_isf/Visf); pH_pc = -log(H_pc/HpNorm); pH_sc = -log(H_sc/HpNorm); pH_isf = -log(H_isf/HpNorm); } // e n d o f c o d e // ASSUMPTIONS & LIMITATIONS: // 1) Circulatory resistances constant, independant of volume (except // Vena cava and systemic arteries). // 2) Time-varying elastances are all chopped sinusoid functions. // 3) Blood viscosity constant, independent of vessel volume. // 4) Vessel walls strictly elastic, not visco-elastic. // 5) Extramural pressure assumed to be zero and constant for circulatory // components. // 6) Resistance of aortic valve equal to Lu et al. tricuspid valve // resistance for default parameter value. // 7) Model is a closed system. // 8) Lx (unquantified inertance element) and Rx (unquantified resistor) // in schematic set to zero. // 9) Governing equation for baroreception based on experiments with // dog carotid sinus only; proximal aortic pressure is used as the // input for the baroreceptor's governing equation. //10) We used a discrete heart rate and contractility function. // Without this function heartbeats get interrupted and become erratic. // Heartrate updates at the end of each cardiac cycle, and when the model // is stabilized, the point of update occurs during the troughs // of the dynamic heart rate and contractility waveforms. //11) The total alveolar volume in the lung is respresented as one // alveolus. //12) Pressure driving breathing is sinusoidal. //13) No spatial component to gas exchange or blood gas handling model. //14) Solubility constants for O2 and CO2 in red blood cells same as in // plasma //15) Kidneys (H+ ion sink function "Hout") act as a high performance, // low pass filter. // COMMENTS: // Disparities between this model and model in source publications: // 1) No aortic valve resistance listed in Lu et al. (2001). // We used a low valve resistance (tricuspid) for this parameter. // Note: Tricuspid resistance listed as "Rta" in Glossary of // publication is not listed in Table 4. // 2) Using the values listed in Table 5 of Lu et al. (2001), we // plotted Pvc vs. Vvc after running the syscirc model for several // cardiac cycles. There was a discontinuity in this graph // when Vvc = Vo (Vena cava volume = unstressed Vena cava // volume). The curve above Vo began at a lower value // than the end of the curve below Vo. We therefore replaced the // D1 parameter in the equation for distended Vena cava volume // with the expression for undistended Vena cava volume solved at // Vo. // 3) No value is given for the distal pulmonary arterial inertance // element Lu et al. (2001) Fig. 1. We assumed this value to be // zero and only used the proximal pulmonary arterial inertance. // The distal inertance element is labeled "Lx" in our schematic. // 4) No value is given for the radial flow resistance of the distal // pulmonary arterial system. This value was also assumed to be // zero and the term was not included in the code. The resistor // is labeled "Rx" in our schematic. // 5) Unlike the Lu et al. model, our heart model does not // include ventricular interaction or a pericardium. // 6) Value of baroreceptor parameter "a" not given an explicit // value in either Lu et al. (2001) or Kezdi and Geller (1967). // A value of 0.001 is assumed as the default. // 7) Our heart model uses a sine activation function while Lu et // al. (2001) use a summation of three gaussian curves. Thus, // we could only calculate bfs_con, but not employ it. Changes // in contractility do not affect our activation function. // 8) Lu et al. (2001) use a previously published heart model (Chung // et al. 1997). Their unstressed end-diastolic and end-systolic // ventricular (and possibly atrial) volumes are static parameters, // while ours can vary with the contraction and relaxation of the heart // chambers. // 9) The signs for the Acw and Bcw parameters are inverted from the // Athanasiades paper. // In our model Pcw = -Acw + Bcw*log((TLC-RV)/(Vcw-RV) - 0.999); // The sign for parameter Bc is inverted from manuscript and the // last else condition (Vc/Vcmax < Vccrit) is ignored. // 10) Lu et al. (2201) do not include an interstitial compartment in // their gas exchange model // 11) We do not include the spatial component requiring partial differential // equations that Lu et al. (2001) use in their gas exchange model. // PARAMETER AND VARIABLE GLOSSARY: // P A R A M E T E R S: // CODEWORD|SYMBOL| NAME | UNITS | DESCRIPTION //----------------------------------------------------------------------------------- // a delta t Time sec Time constant for baroreceptor firing // rate // a_con Dimensionless Time constant for efferent // contractility firing // a_hrs Dimensionless Time constant for efferent sympathetic // heart rate firing // a_hrv Dimensionless Time constant for efferent vagal // firing // a_vaso Dimensionless Time constant for efferent vasomotor // tone firing // a1 delta t Time sec Time constant for baroreceptor firing // rate // a2 delta t Time sec Time constant for baroreceptor firing // rate // Ac P Pressure cmH2O Offset for collapsible airways recoil // pressure // Acw P Pressure cmH2O Offset for chest wall recoil pressure // Al P Pressure cmH2O Lung recoil pressure scaling parameter // alphaCO2 M*mmHg^(-1) Solubility constant of CO2 in plasma // alphaO2 M*mmHg^(-1) Solubility constant of O2 in plasma // amin Dimensionless Contractility control offset // Amus P Pressure cmH20 Respiratory muscle pressure scaling // parameter // Au R Resistance cmH2O*sec/L Offset for upper airways resistance // As R Resistance cmH2O*sec/L Small airways resistance scaling parameter // b_con Dimensionless Time constant for efferent sympathetic // contractility firing // b_hrs Dimensionless Time constant for efferent sympathetic // heart rate firing // b_hrv Dimensionless Time constant for efferent vagal firing // b_vaso Dimensionless Time constant for efferent vasomotor // tone firing // Bc P Pressure cmH2O Collapsible airways recoil pressure // scaling parameter // Bcp P Pressure cmH2O Collapsible airways recoil pressure // scaling parameter // Bcw P Pressure cmH20 Chest wall recoil pressure scaling // parameter // Betab M Molarity M BUFFERING CAPACITY in BLOOD // Betat M Molarity M BUFFERING CAPACITY in Interstitial space // Bl P Pressure cmH20 Offset for lung recoil pressure // Bs R Resistance cmH2O*sec/L Offset for small airways resistance // bmin Dimensionless Contractility control offset // Caod Compliance ml*mmHg^(-1) Aortic distal compliance // Caop Compliance ml*mmHg^(-1) Aortic proximal compliance // CFb Dimensionless CARBONIC ANHYDRASE CATALYTIC FACTOR in BLOOD // CFt Dimensionless CARBONIC ANHYDRASE CATALYTIC FACTOR in ISF // CHb M Molarity M Concentration of O2 or CO2 binding sites // on Hemoglobin // Cheme M Molarity M CONC of hemeHb in BLOOD (4*150/65000 M) // Cpa Compliance ml*mmHg^(-1) Pulmonary arterioles compliance // Cpc Compliance ml*mmHg^(-1) Pulmonary capillaries compliance // Cpv Compliance ml*mmHg^(-1) Pulmonary veins compliance // Csad Compliance ml*mmHg^(-1) Systemic arterioles compliance // Csc Compliance ml*mmHg^(-1) Systemic capillaries compliance // Ctpad Compliance ml*mmHg^(-1) Distal pulmonary arterial compliance // Ctpap Compliance ml*mmHg^(-1) Proximal pulmonary arterial compliance // Cve Compliance L*cmH2O^(-1) Viscoelastic compliance of lung // D1 P Pressure mmHg Offsetting constant for unstressed // and distended Vena cava pressure // D2 P Pressure mmHg Offsetting constant for partially // collapsed Vena cava pressure // Do V Volume ml Active vasomotor tone volume parameter // for systemic arterial pressure // Emaxla Elastance mmHg*ml^(-1) Maximum elastance of left atrium // Emaxlv Elastance mmHg*ml^(-1) Maximum elastance of left ventricle // Emaxra Elastance mmHg*ml^(-1) Maximum elastance of right atrium // Emaxrv Elastance mmHg*ml^(-1) Maximum elastance of right ventricle // Eminla Elastance mmHg*ml^(-1) Minimum elastance of left atrium // Eminlv Elastance mmHg*ml^(-1) Minimum elastance of left ventricle // Eminra Elastance mmHg*ml^(-1) Minimum elastance of right atrium // Eminrv Elastance mmHg*ml^(-1) Minimum elastance of right ventricle // f Frequency min^(-1) Breathing frequency (breaths/min) // facid ratio molar acid produced per oxygen mole // FRC V Volume L Minimum lung volume // H Dimensionless Hematocrit // h1 Frequency min^(-1) Heart rate control offset parameter // h2 Frequency min^(-1) Heart rate control scaling parameter // h3 Frequency min^(-1) Heart rate control scaling parameter // h4 Frequency min^(-1) Heart rate control scaling parameter // h5 Frequency min^(-1) Heart rate control scaling parameter // h6 Frequency min^(-1) Heart rate control scaling parameter // Hbconc g*L^(-1) CONC of Hb in BLOOD (0.15 gm Hb/ml BLOOD) // HpNorm M Molarity M H+ NORMALIZATION VARIABLE // K sec^(-1)*mmHg^(-1) Baroreceptor gain // K_con Dimensionless CNS gain for contractility control // K_hrs Dimensionless CNS gain for sympathetic heart rate // control // K_hrv Dimensionless CNS gain for vagal heart rate control // K_vaso Dimensionless CNS gain for vasomotor tone control // K1 mmHg*ml^(-1) Scaling constant for pressure of // unstressed and distended Vena cava // K1bgh M Molarity M H2CO3 IONIZATION CONST // K3bgh M Molarity M O2Hb AMINO GROUP IONIZATION CONST // K4bgh M^(-1.636) EQUILIBRIUM CONST for O2-Hb BINDinG // K5bgh Dimensionless Hb CARBOMATE IONIZATION CONST // K6bgh Dimensionless O2Hb CARBOMATE IONIZATION CONST // K2 P Pressure mmHg Scaling constant for pressure of // partially collapsed Vena cava // K2bgh M Molarity M Hb AMINO GROUP IONIZATION CONST // Ka Dimensionless Contractility control scaling factor // Kb Dimensionless Contractility control scaling factor // Kc P Pressure mmHg Active vasomotor tone scaling parameter // for systemic arterial pressure // Kc_air R Resistance cmH2O*sec*L^(-1) Collapsible airways resistance scaling // parameter // Kcytox M Molarity M Km for cytochrome oxidase in systemic // interstitial fluid // Kl L^(-1) Lung recoil pressure scaling parameter // km1 Frequency sec^(-1) BACKWARD RATE CONST in CO2+H2O REACTION // kp1 Frequency sec^(-1) FORWARD RATE CONST in CO2+H2O REACTION // Kp1 P Pressure mmHg Passive vasomotor tone scaling parameter // for systemic arterial pressure // Kp2 mmHg*ml^(-2) Passive vasomotor tone scaling parameter // for systemic arterial pressure // kp5 M^(-1)*sec^(-1) FORWARD RATE CONST for Hb CARBOMATE REACTION // Kr R Resistance mmHg*sec*ml^(-1) Pressure scaling constant for systemic // arterial resistance // KR R Resistance mmHg*sec*ml^(-1) Scaling factor for Vena cava resistance // Ks Dimensionless Small airways resistance scaling parameter // Ku cmH2O*sec^2*L^(-2) Upper airways resistance scaling parameter // Kv P Pressure mmHg Scaling factor for systemic venous // pressure // L_con delta t Time sec CNS time delay for contractility control // L_hrs delta t Time sec CNS time delay for sympathetic heart // rate control // L_hrv delta t Time sec CNS time delay for vagal heart rate // control // L_vaso delta t Time sec CNS time delay for vasomotor tone // control // Laod Inertance mmHg*sec^2*ml^(-1) Distal aorta inertance // Laop Inertance mmHg*sec^2*ml^(-1) Proximal aorta inertance // Lpa Inertance mmHg*sec^2*ml^(-1) Pulmonary arterial inertance // nH Dimensionless Hill coefficient for O2-Hemoglobin binding // No_con Frequency sec^(-1) Frequency parameter for efferent // sympathetic contractility firing // No_hrs Frequency sec^(-1) Frequency parameter for efferent // sympathetic heart rate firing // No_hrv Frequency sec^(-1) Frequency parameter for efferent vagal // firing // No_vaso Frequency sec^(-1) Frequency parameter for efferent // vasomotor tone firing // O2cap mol*g^(-1) O2 CAPACITY of Hb (1.34 ml O2/gm Hb) // P50_CO2 P Pressure mmHg P50 of CO2 for Hemoglobin // P50_O2 P Pressure mmHg P50 of O2 for Hemoglobin // Pao P Pressure mmHg External atmospheric pressure // PH2O P Pressure mmHg Vapor pressure of water at body temperature // PRint delta t Time sec Difference in atrial and ventricular // activation times // PS PS Permeability ml*min^(-1) Permeability-surface area product for gas // exchange // PS_H PS Permeability ml*min^(-1)*g^(-1) PS for H+ across CAP Membrane // PS_HCO3 PS Permeability ml*min^(-1)*g^(-1) PS for HCO3- across CAP Membrane // Pstp P Pressure mmHg Standard pressure // Raod R Resistance mmHg*sec*ml^(-1) Distal aortic resistance // Raop R Resistance mmHg*sec*ml^(-1) Proximal aortic resistance // Rav R Resistance mmHg*sec*ml^(-1) Aortic valve resistance // Rcor R Resistance mmHg*sec*ml^(-1) Coronary circulation resistance // Rcrb R Resistance mmHg*sec*ml^(-1) Cerebral circulation resistance // Rla R Resistance mmHg*sec*ml^(-1) Mitral valve resistance // Ro R Resistance mmHg*sec*ml^(-1) Vena cava resistance offset parameter // Rpa R Resistance mmHg*sec*ml^(-1) Pulmonary arterioles resistance // Rpad R Resistance mmHg*sec*ml^(-1) Distal proximal pulmonary resistance // Rpap R Resistance mmHg*sec*ml^(-1) Proximal pulmonary resistance // Rpc R Resistance mmHg*sec*ml^(-1) Pulmonary capillaries resistance // Rps R Resistance mmHg*sec*ml^(-1) Pulmonary shunt resistance // Rpuv R Resistance mmHg*sec*ml^(-1) Pulmonary valve resistance // Rpv R Resistance mmHg*sec*ml^(-1) Pulmonary veins resistance // RQ Dimensionless Respiratory quotient // Rra R Resistance mmHg*sec*ml^(-1) Tricuspid valve resistance // Rsad R Resistance mmHg*sec*ml^(-1) Systemic arteriolar resistance // Rsc R Resistance mmHg*sec*ml^(-1) Systemic capillaries resistance // Rsv R Resistance mmHg*sec*ml^(-1) Systemic veins resistance // Rtaod R Resistance mmHg*sec*ml^(-1) Transmural distal aortic resistance // Rtaop R Resistance mmHg*sec*ml^(-1) Transmural proximal aortic resistance // Rtpap R Resistance mmHg*sec*ml^(-1) Proximal pulmonary arterial transmural // resistance // RV V Volume L Residual volume // Rve R Resistance cmH2O*sec*L^(-1) Resistance of lung viscoelastance // T_con delta t Time sec CNS time parameter for contractility // control // T_hrs delta t Time sec CNS time parameter for sympathetic // heart rate control // T_hrv delta t Time sec CNS time parameter for vagal heart // rate control // T_vaso delta t Time sec CNS time parameter for vasomotor tone // control // tau delta t Time sec Time constant for chest wall volume // follower // tau_con delta t Time sec Time parameter for efferent // sympathetic contractility firing // tau_hrs delta t Time sec Time parameter for efferent // sympathetic heart rate firing // tau_hrv delta t Time sec Time parameter for efferent vagal // firing // tau_p ml^(-1) Passive vasomotor tone constant for // systemic arterial pressure // tau_vaso delta t Time sec Time parameter for efferent // vasomotor tone firing // tauHout delta t Time sec Time constant for H+ removal // Tbody T Temperature degrees Kelvin Body temperature // TLC V Volume L Total lung capacity // Tstp T Temperature degrees Kelvin Standard temperature // tweight W Mass kg Tissue weight of body // Vcmax V Volume L Collapsible airwyas resistance parameter // Vcytox mol*min^(-1) Vmax for cytochrome oxidase in systemic // interstitial fluid // VD V Volume L Lung dead space // Visf V Volume ml Systemic interstitial fluid volume // Vlard V Volume ml Unstressed end-diastolic left atrium // volume // Vlars V Volume ml Unstressed end-systolic left atrium // volume // Vlvrd V Volume ml Unstressed end-diastolic left ventricle // volume // Vlvrs V Volume ml Unstressed end-systolic left ventricle // volume // Vmax_sv V Volume ml Maximal volume of lumped systemic // veins // Vmax_vc V Volume ml Maximum volume of Vena cava // Vmin_vc V Volume ml Mimimum volume of Vena cava // Vo V Volume ml Unstressed volume of Vena cava // Vpcmax V Volume L Maximum pulmonary capillary volume // Vrard V Volume ml Unstressed end-diastolic right ventricle // volume // Vrars V Volume ml Unstressed end-systolic right ventricle // volume // Vrvrd V Volume ml Unstressed end-diastolic right ventricle // volume // Vrvrs V Volume ml Unstressed end-systolic right ventricle // volume // Vsa_max V Volume ml Maximal luminal volume of systemic // arteries // Vsa_o V Volume ml Minimal volume of systemic arteries // Vstar V Volume L Small airways resistance parameter // y0 Dimensionless Cut off for ventricular sine // function (elastances) // ya0 Dimensionless Cut off for atrial sine function // (elastances) //----------------------------------------------------------------------------------- //V A R I A B L E S: // CODEWORD|SYMBOL| NAME | UNITS | DESCRIPTION //----------------------------------------------------------------------------------- // afs_con Dimensionless Dynamic heart contractility scaling // function // afs_con2 Dimensionless Discrete heart contractility scaling // function // B_isf M Molarity M HCO3- ION CONC in SYST ISF // B_pc M Molarity M HCO3- ION CONC in PULM CAP // B_sc M Molarity M HCO3- ION CONC in SYST CAP // bfs_con Dimensionless Contractility variable to alter Lu et // al. (2001) activation function // CO2flux J Flux ml*min^(-1) CO2 flux from alveoli to pulmonary // capillaries // CtCO2_pc M Molarity M Concentration of CO2 in pulmonary // capillaries // CtCO2_sc M Molarity M Concentration of CO2 in systemic // capillaries // CtO2_pc M Molarity M Concentration of O2 in pulmonary // capillaries // CtO2_sc M Molarity M Concentration of O2 in systemic // capillaries // DL_CO2 mlSTPD*s^(-1)*mmHg^(-1) Diffusing capacity of lungs to CO2 // DL_O2 mlSTPD*s^(-1)*mmHg^(-1) Diffusing capacity of lungs to O2 // dSHbCO2dPCO2_pc mmHg^-1 Derivative of SHbCO2 with respect to // PCO2_pc // dSHbCO2dPCO2_sc mmHg^-1 Derivative of SHbCO2 with respect to // PCO2_sc // dSHbO2dPO2_pc mmHg^-1 Derivative of SHbO2 with respect to // PO2_pc // dSHbO2dPO2_sc mmHg^-1 Derivative of SHbO2 with respect to // PCO2_sc // dV V Volume L Chest wall volume follower to get flow // DVRO11 Dimensionless COEFF of dpO11/dt in d(pO11*VRO11)/dt // DVRO21 Dimensionless COEFF of dpO21/dt in d(pO21*VRO21)/dt // Ela Elastance mmHg*ml^(-1) Time-varying elastance of left atrium // Elv Elastance mmHg*ml^(-1) Time-varying elastance of left ventricle // Era Elastance mmHg*ml^(-1) Time-varying elastance of right atrium // Erv Elastance mmHg*ml^(-1) Time-varying elastance of right ventricle // F_hrs Dimensionless Normalized discharge rate for // sympathetic heart rate control // F_hrv Dimensionless Normalized vagal discharge rate // F_vaso Dimensionless Normalized discharge rate for vasomotor // control // Faod F Flow ml*sec^(-1) Distal aorta flow // Faop F Flow ml*sec^(-1) Proximal aorta flow // Fcor F Flow ml*sec^(-1) Coronary flow // Fcrb F Flow ml*sec^(-1) Cerebral flow // Fla F Flow ml*sec^(-1) Flow across mitral valve // Flv F Flow ml*sec^(-1) Flow across aortic valve // Fpa F Flow ml*sec^(-1) Pulmonary arterioles flow // Fpad F Flow ml*sec^(-1) Distal pulmonary arterial flow // Fpap F Flow ml*sec^(-1) Proximal pulmonary arterial flow // Fpc F Flow ml*sec^(-1) Pulmonary capillaries flow // Fps F Flow ml*sec^(-1) Pulmonary shunt flow // Fpv F Flow ml*sec^(-1) Pulmonary veins flow // Fra F Flow ml*sec^(-1) Flow across tricuspid valve // Frv F Flow ml*sec^(-1) Flow across pulmonary valve // Fsa F Flow ml*sec^(-1) Systemic arteries flow // Fsad F Flow ml*sec^(-1) Systemic arterioles flow // Fsc F Flow ml*sec^(-1) Systemic capillaries flow // fs_con Dimensionless Normalized discharge rate for // contractility control // Fsv F Flow ml*sec^(-1) Systemic veins flow // Ftaod F Flow ml*sec^(-1) Distal aorta radial flow // Ftaop F Flow ml*sec^(-1) Proximal aorta radial flow // Ftpa F Flow ml*sec^(-1) Pulmonary arterioles radial flow // Ftpad F Flow ml*sec^(-1) Distal pulmonary arterial radial flow // Ftpap F Flow ml*sec^(-1) Proximal pulmonary arterial radial flow // Ftpc F Flow ml*sec^(-1) Pulmonary capillaries radial flow // Ftpv F Flow ml*sec^(-1) Pulmonary veins radial flow // Ftsa F Flow ml*sec^(-1) Systemic arteries radial flow // Ftsad F Flow ml*sec^(-1) Systemic arterioles radial flow // Ftsc F Flow ml*sec^(-1) Systemic capillaries radial flow // Ftsv F Flow ml*sec^(-1) Systemic veins radial flow // Ftvc F Flow ml*sec^(-1) Vena cava radial flow // Fvc F Flow ml*sec^(-1) Vena cava flow // Fx F Flow ml*sec^(-1) Blood loss out of left ventricle // GO2isf ml*min^(-1)*g^(-1) GULOSITY FOR O2 CONSUMPTION in TISSUE // H_isf M Molarity M H+ ION CONC in SYST ISF // H_pc M Molarity M H+ ION CONC in PULM CAP // H_sc M Molarity M H+ ION CONC in SYST CAP // Hout M Molarity M Variable for H+ ion sink (crude kidney) // HR Frequency min^(-1) Dynamic heart rate // HR2 Frequency min^(-1) Discrete heart rate function // N_con Frequency sec^(-1) Sympathetic discharge rate at CNS for // contractility // N_hrs Frequency sec^(-1) Sympathetic discharge rate at CNS // controlling heart rate // N_hrv Frequency sec^(-1) Vagal discharge rate at CNS controlling // heart rate // N_vaso Frequency sec^(-1) Sympathetic discharge rate at CNS // controlling vasomotor tone // Nbr Frequency sec^(-1) Instantaneous firing frequency of // baroreceptor // Nbr_t sec^(-2) Time derivative of baroreceptor firing // rate // O2flux F Flow ml*min^(-1) O2 flux from alveoli to pulmonary // capillaries // PA P Pressure cmH2O Alveolar pressure // PA_CO2 P Pressure mmHg Partial pressure of CO2 in alveoli // PA_N2 P Pressure mmHg Partial pressure of N2 in alveoli // PA_O2 P Pressure mmHg Partial pressure of O2 in alveoli // Pao_CO2 P Pressure mmHg Partial pressure of CO2 at airway opening // Pao_N2 P Pressure mmHg Partial pressure of N2 at airway opening // Pao_O2 P Pressure mmHg Partial pressure of O2 at airway opening // Paod P Pressure mmHg Distal aorta pressure // Paop P Pressure mmHg Proximal aorta pressure // Paop1 P Pressure mmHg Dummy variable for proximal aorta // pressure // Paop2 P Pressure mmHg Dummy variable for distal aorta pressure // PBC_pc M Molarity M HbNHCOO- ION CONC in PULM CAP // PBC_sc M Molarity M HbNHCOO- ION CONC in SYST CAP // Pc P Pressure cmH2O Collapsible airways pressure // PC_CO2 P Pressure mmHg Partial pressure of CO2 in collapsible // airways // PC_N2 P Pressure mmHg Partial pressure of N2 in collapsible // airways // PC_O2 P Pressure mmHg Partial pressure of O2 in collapsible // airways // PCO2_pc P Pressure mmHg Partial pressure of CO2 in pulmonary // capillaries // PCO2_sc P Pressure mmHg Partial pressure of CO2 in systemic // capillaries // PCO2_isf P Pressure mmHg Partial pressure of CO2 in systemic // interstitial fluid // PD P Pressure cmH2O Total rigid dead space pressure // PD_CO2 P Pressure mmHg Partial pressure of CO2 in lung dead space // PD_N2 P Pressure mmHg Partial pressure of N2 in lung dead space // PD_O2 P Pressure mmHg Partial pressure of O2 in lung dead space // Pcw P Pressure cmH2O Chest wall recoil pressure // pH_isf Dimensionless pH in SYST ISF // pH_pc Dimensionless pH in PULM CAP // pH_sc Dimensionless pH in SYST CAP // phi Dimensionless Phase shift of discrete heart rate function // Pl P Pressure cmH2O Lung recoil pressure // Pla P Pressure mmHg Left atrial pressure // Plv P Pressure mmHg Left ventricular pressure // Pmus P Pressure cmH2O Applied respiratory muscle pressure // PO2_isf P Pressure mmHg Partial pressure of O2 in systemic // interstitial fluid // PO2_pc P Pressure mmHg Partial pressure of O2 in pulmonary // capillaries // PO2_sc P Pressure mmHg Partial pressure of O2 in systemic // capillaries // Ppa P Pressure mmHg Pulmonary arterioles pressure // Ppad P Pressure mmHg Distal pulmonary arterial pressure // Ppap P Pressure mmHg Proximal pulmonary arterial pressure // Ppap1 P Pressure mmHg Dummy variable for Ppap // Ppap2 P Pressure mmHg Dummy variable for Ppap // Ppc P Pressure mmHg Pulmonary capillaries pressure // Ppl P Pressure cmH2O Pleural pressure // Ppv P Pressure mmHg Pulmonary veins pressure // Pra P Pressure mmHg Right atrial pressure // Prv P Pressure mmHg Right ventricular pressure // Psa P Pressure mmHg Systemic arterial pressure // Psa_a P Pressure mmHg Active systemic arterial pressure component // Psa_p P Pressure mmHg Passive systemic arterial pressure component // Psad P Pressure mmHg Systemic arterioles pressure // Psc P Pressure mmHg Systemic capillaries pressure // Psv P Pressure mmHg Systemic veins pressure // Pvc P Pressure mmHg Vena cava pressure // Pve P Pressure cmH2O Viscoelastic lung pressure // QdotCA F Flow ml*s^(-1) Flow through small airways // QdotDC F Flow ml*s^(-1) Flow through collapsible airways // QdotED F Flow ml*s^(-1) Flow through upper airways // Rc R Resistance cmH2O*sec*L^(-1) Collapsible airways resistance // Rs R Resistance cmH2O*sec*L^(-1) Small airways resistance // Rsa R Resistance mmHg*sec*ml^(-1) Resistance of systemic arteries // Ru R Resistance cmH2O*sec*L^(-1) Upper airways resistance // Rvc R Resistance mmHg*sec*ml^(-1) Resistance of Vena cava // SHbCO2_pc Dimensionless CO2 saturation of Hemoglobin in // pulmonary capillaries // SHbCO2_sc Dimensionless CO2 saturation of Hemoglobin in // systemic capillaries // SHbO2_pc Dimensionless Oxygen saturation of Hemoglobin in // pulmonary capillaries // SHbO2_sc Dimensionless Oxygen saturation of Hemoglobin in // systemic capillaries // tshift delta t Time sec Time shift of discrete heart rate function // V_CO2 mole*min^(-1) CO2 production rate in systemic tissue // V_O2 mole*min^(-1) O2 consumption rate in systemic tissue // VA V Volume L Alveolar volume // Vaod V Volume ml Distal aorta volume // Vaop V Volume ml Proximal aorta volume // Vc V Volume L Volume of collapsible airways // Vcw V Volume L Chest wall volume // Vla V Volume ml Volume of left atrium // Vlar V Volume ml Unstressed left atrial volume // Vlv V Volume ml Volume of left ventricle // Vlvr V Volume ml Unstressed left ventricular volume // Vpa V Volume ml Pulmonary arterioles volume // Vpad V Volume ml Distal pulmonary arterial volume // Vpap V Volume ml Proximal pulmonary arterial volume // Vpc V Volume ml Pulmonary capillaries volume // Vpv V Volume ml Pulmonary veins volume // Vra V Volume ml Volume of right atrium // Vrar V Volume ml Unstressed right atrial volume // Vrv V Volume ml Volume of right ventricle // Vrvr V Volume ml Unstressed right ventricular volume // Vsa V Volume ml Systemic arteries volume // Vsad V Volume ml Systemic arterioles volume // Vsc V Volume ml Systemic capillaries volume // Vsv V Volume ml Systemic veins volume // Vvc V Volume ml Vena cava volume // Vve V Volume L Lung viscoelastic volume // y Dimensionless Sine wave for ventricle elastance // (period = 1/HR2) // ya Dimensionless Sine wave for atrial elastance // (period = 1/HR2) //----------------------------------------------------------------------------------- // SCHEMATICS // 1) Circulation: // Fps // | Rps // Lx : \ // | 0----->-------^^^^^-------------0 // | | | // | | Fpa Fpc | Ppv // | Ppa | Rpa | Ppc Rpc | | / // : \| \ : / \ : |/ // 0-----CCCCC-->--0-----^^^^^-->--0-----^^^^^-->--0------------0 // | / | | | | // | Fpad ----- Cpa, ----- Cpc, ----- Cpv, | // > ----- Vpa ----- Vpc ----- Vpv | // Ctpad, > - Rpad | | | | // Vpad > v - Fpa v - Fpc v - Fpv | // Rx \ | /// /// /// > // / \ || | > - Rpv // /--<--^^^^^--||--0 - Ppad > // / : || | | // | ^ - Fpap | // | | | // C v - Fpv // Ftpad Lpa - C | // C | // | | // | | // 0 | // | | // | / // > ------------------------- // Ctvap, > - Rpap / // Vpap > | // Rtpap \ | | // / \ || | | // /-<--^^^^^--||--0 - Ppap | // / : || | | // | ^ - Frv | // | | | // Ftpap > | // > - Rpuv | // > | // | | // | Pulmonary valve Ela, | // _ / Vla | // ^ \ | // * | || / / || | * // *Prv* - 0-----VE----/ /----VE------0 - *Pla* // * | || / / || | * // | / | // | Erv, | // | Vrv | /// // Rra | Mitral valve | ^ - Ftaop // \ | \ | | // -->|-^^^^^- --^^^^^-|<-- > // | \ | \ > - Rtaop // | Tricuspid valve | Rla > // | | | // | | ----- - Caop // | Era, Elv, | ----- // | Vra Vlv * | Rav Paop | Raop // | / \ *Plv*| | | | | // * | || / / || * \| : : | : // *Pra* - 0-----VE----/ /----VE-----0-->|--^^^^^--0-----0--^^^^^--0 // * | || / / || / | | // | / | | // | Aortic valve | | // | | v - Faop // 0--------------------^^^^^-----------<----------0 | // | | | | | // | Rcor Fcor | C // | | Laop - C // | -^^^^^-----------<----------0 C Caod // | / | | | / Rtaod // | / Rcrb Fcrb | || | / // | / Paod --- 0--||--^^^^^-->-/ // | / | || / / // | / | Ftaod // | / > // > / > - Raod // > - Rvc / > // > / | // | / | // | / 0 // | / | // | / | // | / C // | / Psv Psc Psad C - Laod // | / Rsv | Rsc | Rsad | Rsa C // | / | | | | | | | | // Fvc - ^ / | Fsv | | Fsc | | Fsad | | Fsa v - Faod // |/ : \ : : \ : : \ : : \ | // Pvc - 0----^^^^^---<--0----^^^^^---<--0----^^^^^---<--0----^^^^^---<--0 - Psa // | | | | | // ----- ----- Csv, ----- Csc, ----- Csad, ----- Vsa // ----- Vvc ----- Vsv ----- Vsc ----- Vsad ----- // | | | | | // v - Ftvc v - Ftsv v - Ftsc v - Ftsad v - Ftsa // /// /// /// /// /// // 2) Airway mechanics (reproduced from Athanasiades (2000) fig. 1B.) // / // o------/ (atm) o // | / | // | | // > | // > Ru | // > | // | | // | | // o | // | | // | | // > | // > Rc | // > Vc | // | / | // | || | // o-------||---------------------o | // | +||- | Vcw | // | | \ | // > | || /---\ | / // > Rs o-----||----| P |--o-----/ (atm) // > Rve | +||- -\---/+ | / // | o---^^^^^---o | / | // | +||- | | | Pmus | // o-------||----o o----o o // || | || | // / o-----||----o // VA || // / // Cve, Vve // LEGEND: // // C ^ || // C or CCCCC = Inertance < > = direction of flow || or ----- = Compliance // C v || ----- // // > / // 0 = Junction ^^^^^ or > = Resistance /// or / = Ground // > / // // _ || /---\ // >| or |< or ^ = diode (1-way valve) VE = Varying elastance | P | = Pressure driver // || \---/ // // : = indicates a labeling arrow, not a path of flow // PUBLISHED PARAMETER VALUES // // This is a record of the original parameter // values from the literature. If the model is // run with these values, it will likely crash. // Vlvrd = 30 // Vlvrs = 10 // Vrvrd = 30 // Vrvrs = 10 // Vlard = 30 // Vlars = 10 // Vrard = 30 // Vrars = 10 // Rra = 0.05 // Rla = 0.01 // PRint = 0.05 // Emaxlv = 2.5 // Eminlv = 0.06 // Emaxrv = 1.2 // Eminrv = 0.04 // Emaxra = 1.2 // Eminra = 0.06 // Emaxla = 1.2 // Eminla = 0.06 // y0 = 0 // ya0 = 0 // Rav = 0.015 // Raop = 0.005 // Rtaop = 0.06 // Rcor = 30 // Rcrb = 10 // Raod = 0.015 // Rtaod = 0.0125 // Rsad = 0.8 // Rsc = 0.6 // Rsv = 0.17 // Caop = 1.6 // Caod = 0.2 // Csad = 0.069 // Csc = 0.2 // Laop = 0.01226 // Laod = 0.0017 // Kc = 1000 // Do = 50 // Vsa_o = 210 // Vsa_max = 250 // Kp1 = 0.03 // Kp2 = 0.2 // Kr = 0.04 // tau_p = 0.1 // Kv = 40 // Vmax_sv = 3500 // D1 = 0 // D2 = -5 // K1 = 0.15 // K2 = 0.4 // KR = 0.001 // Ro = 0.025 // Vo = 130 // Vmax_vc = 350 // Vmin_vc = 50 // Rpuv = 0.015 // Rtpap = 0.02 // Rpap = 0.002 // Rpad = 0.005 // Rps = 4.5 // Rpa = 0.005 // Rpc = 0.008 // Rpv = 0.008 // Ctpap = 0.8 // Ctpad = 0.05 // Cpa = 0.5 // Cpc = 1 // Cpv = 4 // Lpa = 1.8E-4 // a = 0.001 // a1 = 0.036 // a2 = 0.0018 // K = 1 // K_hrv = 0.8 // T_hrv = 1.8 // L_hrv = 0.2 // a_hrv = 0 // b_hrv = 1 // tau_hrv = -0.04 // No_hrv = 110 // K_hrs = 1 // T_hrs = 10 // L_hrs = 3 // a_hrs = 0.3 // b_hrs = 0.7 // tau_hrs = 0.09 // No_hrs = 100 // K_con = 1 // T_con = 10 // L_con = 3 // a_con = 0.3 // b_con = 0.7 // tau_con = 0.04 // No_con = 110 // K_vaso = 1 // T_vaso = 6 // L_vaso = 3 // a_vaso = 0.3 // b_vaso = 0.7 // tau_vaso = 0.04 // No_vaso = 110 // h1 = 35 // h2 = 140 // h3 = 40 // h4 = 32 // h5 = 10 // h6 = 20 // amin = -2 // bmin = 0.7 // Ka = 5 // Kb = 0.5 // FRC = 3.9 // VT = 0.5 // RV = 1.65 // TLC = 5.55 // VD = 0.185 // f = 10 // Ac = 7.09 // Acw = 1.4 // Al = 0.2 // As = 2.2 // Au = 0.34 // Bc = 37.3 // Bcp = 3.73 // Bcw = -3.5 // Bl = -0.5 // Bs = 0.02 // Cve = 0.5 // Kc_air = 0.21 // Kl = 1 // Ks = -10.9 // Ku = 0.46 // Rve = 1 // Vstar = 5.3 // Vcmax = 0.185 // Amus = 2 // tau = 0.1 // Tbody = 300 // Pstp = 760 // Tstp = 273 // nH = 3.5 // P50_O2 = 26.5 // CHb = 0.0204 // H = 0.45 // alphaO2 = 1.36E-6 // Visf = 48000 // PS = 10 // Vcytox = 2.1E-7 // Kcytox = 1E-9 // P50_CO2 = 250 // alphaCO2 = 3.264E-5 // RQ = 0.8 // PH2O = 47 // Vpcmax = 0.07125 // kp1 = 0.12 // km1 = 89 // K1bgh = 5.5E-4 // K2bgh = 1E-7 // K3bgh = 4E-7 // K4bgh = 55250 // K5bgh = 2E-4 // K6bgh = 5E-5 // kp5 = 5000 // CFb = 6000 // CFt = 5000 // Betab = 0.036 // Betat = 0.024 // O2cap = 5.982142857142858E-5 // Hbconc = 150 // Cheme = 0.00923 // HpNorm = 1 // facid = 0.01 // tweight = 60 // tauHout = 0.1 // PS_H = 20 // PS_HCO3 = 20 // Vra__init = 90 // Vrv__init = 130 // Vla__init = 90 // Vlv__init = 130 // Faop__init = 75.3 // Faod__init = 70.6 // Vaop__init = 100 // Vaod__init = 200 // Vsa__init = 225 // Vsad__init = 174 // Vsc__init = 300 // Vsv__init = 3000 // Vvc__init = 240 // Vpap__init = 18 // Vpad__init = 35 // Vpa__init = 75 // Vpc__init = 110 // Vpv__init = 200 // Fpap__init = 67.5 // Nbr__init = 175 // Nbr_t__init = 0 // N_hrv__init = 75 // N_hrs__init = 90 // N_con__init = 90 // N_vaso__init = 90 // Vc__init = 0.15 // VA__init = 1.8 // Vve__init = 0.001 // dV__init = 2 // PO2_isf__init = 25 // PO2_pc__init = 100 // PO2_sc__init = 40 // PCO2_pc__init = 40 // PCO2_sc__init = 50 // PCO2_isf__init = 60 // PD_O2__init = 100 // PC_O2__init = 100 // PA_O2__init = 100 // PD_CO2__init = 35 // PC_CO2__init = 35 // PA_CO2__init = 35 // PBC_pc__init = 0.00235 // PBC_sc__init = 0.00235 // B_pc__init = 0.0236 // B_sc__init = 0.0236 // B_isf__init = 0.0182 // H_pc__init = 4E-8 // H_sc__init = 4E-8 // H_isf__init = 6.8E-8 // Hout__init = 6.8E-8 // Pao = 760 // r_Pao_O2 = 0.21 // r_Pao_CO2 = 3E-4 // Fx = 0 // Hin = 0 // REFERENCES: // Athanasiades A, Ghorbel F, Clark JW, Niranjan SC, Olansen J, // Zwischenberger JB, Bidani A. 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