Systems Models of the Circula4on BENG 230C Lecture 2

Transcription

1 Systems Models of the Circula4on BENG 230C Lecture 2 Why modeling Enhance insight in physiology Hypothesis genera5on Clinical applica5ons diagnosis training pla7orms for surgeons predict outcomes of surgical interven5ons predict outcomes of therapies

2 Example: Strain in failing hearts Circulatory models

3 Circula5on model elements Compliant vessel qin [ml/s] V [ml] qout [ml/s] Conservation of mass: dv/dt = qin - qout 5 Circula5on model elements Compliant vessel V [ml] p [kpa] C [ml/kpa] Pressure in vessel: p = V/C 6

4 Circula5on model elements Resistance R [kpa*s/ml] pin [kpa] pout [kpa] q [ml/s] Flow over resistance: q = (pin - pout)/r 7 Circula5on model elements Bifurcation q2 q1 q3 Conservation of mass: q1 = q2 + q3 8

5 Circula5on model elements Compliant vessel + Resistance pin qin V R pout qout C pin = V/C qout = (pin-pout)/r dv/dt = qin - qout 9 Circula5on model elements Compliant vessel + Resistance pin qin V R pout qout C By substitution of pin and qout in dv/dt: qin = C*dpin/dt + (pin-pout)/r With: pout = 0 pin = plv qin = qao R = Rper C = Cart this leads to... 10

6 Two- element windkessel model (ODo Frank, 1899) Compliant vessel + Resistance plv qao V Rper qout Cart qao = Cart*dplv/dt + plv/rper 11 windkessel models windkessel = airkettle used in plunger pumps to ensure a steady flow

7 Two- element windkessel model Electric equivalent plv qao V Rper qout Cart I 1 qao I 3 qout U plv I 2 dv/dt Cart Rper 13 Two- element windkessel model Electric equivalent plv qao V Rper qout Cart The 2wk model produces unrealistic aortic pressure waves, especially at higher frequencies 14

8 Three- element windkessel model (Broemser, 1930) Compliant vessel + two Resistances plv qao Zao V part Rper qart Cart p art = V/C art q ao = (p lv -p art )/Z ao q art = p art /R per dv/dt = q ao - q art p lv : LV pressure p art : arterial pressure q ao : aortic flow Z ao : aortic impedance R per : peripheral resistance C art : arterial compliance V: volume of aorta 15 Three- element windkessel model (Broemser, 1930) Compliant vessel + two Resistances plv qao Zao V part Rper qart Cart Produces realistic aortic pressure and flow waves 16

9 Circula5on model elements The LeO Ventricle Time-varying Compliant Cavity Vlv [ml] plv [kpa] Pressure in left ventricle: plv = Vlv/Clv(t) Clv(t) [ml/kpa] Taking into account that V 0 when p=0: plv = (Vlv-Vlv,0)/Clv(t) Or, more commonly, with time-varying elastance E=1/C: plv = Elv(t)*(Vlv-Vlv,0) 17 Circula5on model elements The LeO Ventricle Time-varying Elastance Vlv [ml] plv [kpa] Clv(t) [ml/kpa] We only need to find a function that describes E(t): During diastole: E = Emin At peak systole: E = Emax E(t) = y(t)*(emax - Emin) + Emin When t<dur_systole: y(t) = -cos(2*pi*t/dur_systole)/ else: y(t) = 0 18

10 Circula5on model elements Simple Valve (open/closed) q pin R pout Flow over valve: q = (pin - pout)/r when pin>pout q = 0 when pin<pout 19 Circula5on model elements The LeO Ventricle Time-varying compliant cavity with valves pla qmitral Rmitral Vlv [ml] plv [kpa] Zaorta paorta qaorta Elv(t) [kpa/ml] plv = Elv(t)*(Vlv-Vlv,0) qmitral = (pla - plv)/rmitral qmitral = 0 when pla>plv when pla<plv qaorta = (plv - paorta)/zaorta when plv>paorta qaorta = 0 dvlv/dt = qmitral - qaorta when plv<paorta 20

11 A simple closed-loop circulatory model p lv q ao p art V 1,C 1 Z ao q mit q art R 1 V 2,C 2 R 2 pvein LV connected to 2 compliant vessels: a total of 3 compartments A simple closed-loop circulatory model p lv q ao p art V 1,C 1 Z ao q mit q art R 1 V 2,C 2 1 p art = V 1 /C 1 R 2 if p lv > p art q ao = (p lv (t) - p art )/Z ao else q ao = 0 q art = (p art -p vein )/R 1 dv 1 /dt = q ao - q art p vein 2 p vein = V 2 /C 2 if p vein > p lv q mit = (p vein -p lv (t))/r 2 else q mit = 0 dv 2 /dt = q art - q mit 3 dv LV /dt = q mit - q ao

12 A simple closed-loop circulatory model This leads to a set of 3 ordinary differen5al equa5ons (ODEs): dv 1 /dt = q ao - q art dv 2 /dt = q art - q mit dv LV /dt = q mit - q ao These can be solved numerically using an ODE- solver (e.g. in Matlab) 23 A more detailed closed- loop circulatory model, Circulation model by Lu et al (AJP 2001)

13 Using Matlab ode23 func5on to solve ODEs [t,statevars]=ode23( your_odes,5mespan,y0,op5ons,parameters); input 5mespan: [0 t_end] or [0:dt:t_end] y0: array with ini5al condi5ons op5ons = odeset( RelTol,1e- 4, AbsTol,1e- 6); parameters: array with model parameter values name of func5on your_odes with system of ODEs output: t: 5me array statevars: array with solu5on of state variables Call ode23 from a main program Using Matlab your_odes create M- file your_odes.m with in it: func5on [dy,variables]=your_odes(t,statevars,flag,parameters) your equa5ons go here Variable names same as previous slide, and ignore flag

14 Example of 3- element windkessel model Program wk3_main: % wk3_main dt = 0.01; % 5me step [sec] t_end = 0.6; % end 5me of simula5on [sec] q lv Zao = 7e- 3; % aor5c impedance [kpa*sec/ml] p art Cao = 32.5; % aor5c compliance [ml/kpa] Rper = 0.25; % peripheral resistance [kpa*sec/ml] p lv Z ao % ini5al condi5ons y0(1,1) = 300; % ini5al aor5c volume [ml] % put parameters in array parameters(1) = Zao; parameters(2) = Cao; parameters(3) = Rper; Cao V R per q art op5ons = odeset('reltol',1e- 4,'AbsTol',1e- 3); % Solve the system of ODEs [t,statevars]=ode23( wk3_ode,[0:dt:t_end],y0,op5ons,parameters); % AOer solu5on is obtained, get rest of (non- state) variables values: for i=1:length(t), [dummy,variables(i,:)]=wk3_ode(t(i),statevars(i,:),0,parameters); end Example of 3- element windkessel model Program wk3_ode: func5on [dy,variables]=wk3_ode(t, statevars, flag, parameters) % Retrieve state variables Vao = statevars(1); % Retrieve parameter values Rao=parameters(1); Cao=parameters(2); Rper=parameters(3); % Make up some left ventricular pressure pmax = 16; %[kpa] plv=pmax*(-cos(2*pi*t/0.6)/2+0.5); part = Vao/Cao; if plv > part, qlv = (plv-part)/rao; % ventricular outflow else qlv = 0.0; % one-way valve end qart = part/rper; % arterial flow dy(1,1) = qlv-qart; %dvao/dt % Store non-state variables in array variables, so we have access and can plot these too variables(1) = qlv; variables(2) = qart; variables(3) = plv; variables(4) = part;

15 Example of 3- element windkessel model: results figure(1); clf subplot(311) plot(t,variables(:,[1 2])); ylabel('flows [ml/sec]','fontsize',16); legend('aortic','arterial','fontsize',16); subplot(312) plot(t,variables(:,[3 4])); ylabel('pressures [kpa]','fontsize',16); legend('lv','aortic'); subplot(313) plot(t,statevars(:,1)) xlabel('time [sec]','fontsize',16); ylabel('aortic volume [ml]','fontsize',16); Homework assignment Design and solve this model of the circula5on in Matlab Rap Cap Cvp Rvs Rvp Era Rtric Rpa Rao Ela Rmit Cvs Erv Elv Cas Ras

16 Parameter values % ====================== solution control parameters =============== dt = 0.001; % time step [sec] tend = 3.000; % end time of simulation [sec] % ======================= Heart ===================================== bcl = 0.600; % Basic cycle length [sec] twitchperiod = 0.300; % Duration of systole [sec], so here systole is half the time of the cycle % Left atrium t_av = 0.120; Emaxla = 0.078; Eminla = 0.071; restvlad = 14; % atrioventricular activation delay % maximum elastance of the left atrium % minimum elastance of left atrium % unloaded volume of left atrium % Right atrium Emaxra = 0.03; Eminra = 0.027; restvrad = 14; % maximum elastance of the left atrium % minimum elastance of the left atrium % unloaded volume of left atrium % Left ventricle Emaxlv = 1.5; % kpa/ml; Eminlv = 1/11; % kpa/ml; restvlvd = 26.1; % ml; % Maximum elastance of left ventricle % Minimum elastance of left ventricle % unloaded volume of left ventricle % Right ventricle Emaxrv = 1/3.01; % kpa/ml; Eminrv = 1/32.3; % kpa/ml; restvrvd = 22.3; % ml; % Maximum elastance of right ventricle % Minimum elastance of right ventricle % unloaded volume of right ventricle Parameter values (Cont d) and ini5al condi5ons % ==================== Systemic circulation ========================= R1s = 0.247; % kpa*sec/ml; % Resistance of vessel 1 R2s = 0.051; % kpa*sec/ml; % Resistance of vessel 2 Rlv = 7e-3; % kpa*sec/ml; % Aortic impedance Rmit = 5e-4; % kpa*sec/ml; % Mitral valve resistance C1s = 32.5; % ml/kpa; % Compliance of vessel 1 C2s = 432; % ml/kpa; % Compliance of vessel 2 % ======================= Pulmonary circulation ====================== R1p = 0.005; % kpa*sec/ml; % Resistance of vessel 1 R2p = 0.005; % kpa*sec/ml; % Resistance of vessel 2 Rrv = 2e-3; % kpa*sec/ml; % Arterial impedance of pulmonary artery Rtric = 5e-4; % kpa*sec/ml; % Resistance of tricuspid valve C1p = 41.7; % ml/kpa; % Compliance of vessel 1 C2p = 50; % ml/kpa; % Compliance of vessel 2 % =========================== Initial conditions ====================== i=1; statevar_init(i,1)=45.4;i=i+1; % LV statevar_init(i,1)=352;i=i+1;! % systemic arterial volume statevar_init(i,1)=1.04e3;i=i+1; % systemic venous volume statevar_init(i,1)=35;i=i+1; % RA statevar_init(i,1)=34.7;i=i+1; % RV statevar_init(i,1)=84.9;i=i+1; % pulmonary arterial volume statevar_init(i,1)=93.0;i=i+1; % pulmonary venous volume statevar_init(i,1)=37.3;i=i+1; % LA

17 Parameter values (Cont d) and ini5al condi5ons % ============= Put parameters together in one array ============== i=1; parameters(i)=bcl;i=i+1; parameters(i)=twitchperiod;i=i+1; parameters(i)=t_av;i=i+1; parameters(i)=emaxla;i=i+1; parameters(i)=eminla;i=i+1; parameters(i)=restvlad;i=i+1; parameters(i)=emaxra;i=i+1; parameters(i)=eminra;i=i+1; parameters(i)=restvrad;i=i+1; parameters(i)=r1s;i=i+1; parameters(i)=r2s;i=i+1; parameters(i)=rlv;i=i+1; parameters(i)=rmit;i=i+1; parameters(i)=c1s;i=i+1; parameters(i)=c2s;i=i+1; parameters(i)=restvlvd;i=i+1; parameters(i)=emaxlv;i=i+1; parameters(i)=eminlv;i=i+1; parameters(i)=r1p;i=i+1; parameters(i)=r2p;i=i+1; parameters(i)=rrv;i=i+1; parameters(i)=rtric;i=i+1; parameters(i)=c1p;i=i+1; parameters(i)=c2p;i=i+1; parameters(i)=restvrvd;i=i+1; parameters(i)=emaxrv;i=i+1; parameters(i)=eminrv;i=i+1; options = odeset('reltol',1e-4,'abstol',1e-6); % Solve the system of Ordinary Differential Equations [t,statevars]=ode23('circulation_statevardot',[0:dt:tend],statevar_init,options,parameters); % After solution is obtained, get rest of (non-state) variables values: for i=1:length(t), [dummy,variables(i,:)]=circulation_statevardot(t(i),statevars(i,:),0,parameters); end Programming the ODEs in a different file Program circula-on_statevardot.m function [dy,variables]=circulation_statevardot(t, statevar, flag, parameters) % Retrieve state variables i=1; Vlv = statevar(i); i=i+1; V1s = statevar(i); i=i+1; V2s = statevar(i); i=i+1; Vra = statevar(i); i=i+1; Vrv = statevar(i); i=i+1; V1p = statevar(i); i=i+1; V2p = statevar(i); i=i+1; Vla = statevar(i); i=i+1; % Retrieve parameter values i=1; bcl=parameters(i);i=i+1; twitchperiod=parameters(i);i=i+1; t_av=parameters(i);i=i+1; Emaxla=parameters(i);i=i+1; Eminla=parameters(i);i=i+1; restvlad=parameters(i);i=i+1; Emaxra=parameters(i);i=i+1; Eminra=parameters(i);i=i+1; restvrad=parameters(i);i=i+1; R1s=parameters(i);i=i+1; R2s=parameters(i);i=i+1; Rlv=parameters(i);i=i+1; Rmit=parameters(i);i=i+1; C1s=parameters(i);i=i+1; C2s=parameters(i);i=i+1; restvlvd=parameters(i);i=i+1; Emaxlv=parameters(i);i=i+1; Eminlv=parameters(i);i=i+1; R1p=parameters(i);i=i+1; R2p=parameters(i);i=i+1; Rrv=parameters(i);i=i+1; Rtric=parameters(i);i=i+1; C1p=parameters(i);i=i+1; C2p=parameters(i);i=i+1; restvrvd=parameters(i);i=i+1; Emaxrv=parameters(i);i=i+1; Eminrv=parameters(i);i=i+1; 34

18 ODEs con5nued % ======================= Equations ========================= % ==================== Cardiac elastance ======================= % ========================= Atria ============================ time_a = mod(t+t_av,bcl); % Time for atria, reset to zero for each new heartbeat % Systolic atrial activation function if (time_a<twitchperiod), ya = -cos(2.0*pi*time_a/twitchperiod)/ ;% systole else ya = 0.0; % diastole end % Left atrium Ela = ya*(emaxla-eminla) + Eminla; Pla = Ela*(Vla-restVlad); % Right atrium... right atrial elastance equations go here... % ========================== Ventricles ========================... ventricular elastance plus rest of circulation equations go here More specifically (1) Solve the model from t=0 to t=3 sec in Matlab with ode23 Use subplot to plot the following as a func5on of 5me in figure 1: Plot LV and RV pressures in subplot(311) Plot LV and RV volumes in subplot(312) Plot LV and RV in- and out- flows in subplot(313) In figure 2, plot the pressure- volume diagram for the LV (for all 5 beats) What is the stroke volume and ejec5on frac5on for the last beat?

19 More specifically (2) Next, re- run the model, but at t=3 sec we ll induce an acute leo ventricular infarct. Simulate this by reducing LV maximum elastance by 30%. Stop the simula5on at t=10 sec. In figure 3, plot the pressure- volume diagram for the LV (for all beats from t=0 to t=10 sec). What is the stroke volume and ejec5on frac5on for the last beat? Discuss what you observe. Remember to label your plots appropriately Hand in on paper (due date Tuesday April 12th, 2011) Matlab scripts (40 points) 3 figures (10 points each, total 30) Stroke volumes and ejec5on frac5ons for the normal and infarcted leo ventricle (5 points each, total 20) Discussion for figure 3 (10 points) Also your Matlab scripts to Amran (aasadi@ucsd.edu)