Systems models of the circulation BENG 230C Lecture 2

1
Systems models of the circulationBENG
230CLecture 2

• Roy Kerckhoffs, PhD
• Cardiac Mechanics Research Group
• Dept of Bioengineering,
• University of California, San Diego
• roy_at_bioeng.ucsd.edu

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A simple systems model of the circulation
Ventricular time-varying elastance
RV
LV
3
Using Matlabode23 function to solve ODEs

• t,statevarsode23(your_ODEs,timespan,y0,option
  s,parameters)
• input
• timespan 0 t_end or 0dtt_end
• y0 array with initial conditions
• options odeset(RelTol,1e-4,AbsTol,1e-6)
• parameters array with model parameter values
• name of function your_ODEs with system of ODEs
• output
• t time array
• statevars array with solution of state variables
• Call ode23 from a main program

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Using Matlabyour_ODEs

• create M-file your_ODEs.m with in itfunction
  dy,variablesyour_ODEs(t,statevars,flag,paramete
  rs) your equations go here
• Array names same as previous slide, and ignore
  flag

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Example of 3-element windkessel model

• Program wk3_main
• wk3_main
• dt 0.01 time step sec
• t_end 0.6 end time of simulation sec
• Zao 7e-3 aortic impedance kPasec/ml
• Cao 32.5 aortic compliance ml/kPa
• Rper 0.25 peripheral resistance kPasec/ml
• initial conditions
• y0(1,1) 300 initial aortic volume ml
• put parameters in array
• parameters(1) Zao
• parameters(2) Cao
• parameters(3) Rper
• options odeset('RelTol',1e-4,'AbsTol',1e-3)
• Solve the system of ODEs
• t,statevarsode23(wk3_ODE,0dtt_end,y0,opti
  ons,parameters)

qart
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Example of 3-element windkessel model

• Program wk3_ODE
• function dy,variableswk3_ODE(t, statevars,
  flag, parameters)
• Retrieve state variables
• Vao statevars(1)
• Retrieve parameter values
• Raoparameters(1)
• Caoparameters(2)
• Rperparameters(3)
• Make up some left ventricular pressure
• pmax 16 kPa
• plvpmax(-cos(2pit/0.6)/20.5)
• part Vao/Cao
• if plv gt part,
• qlv (plv-part)/Rao ventricular
  outflow
• else
• qlv 0.0
  one-way valve

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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)

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Homework assignmentDesign and solve this model
of the canine circulation in Matlab
Ventricular time-varying elastance
RV
LV
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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 atrioventricular activation
  delay
• Emaxla 0.078 maximum elastance of the
  left atrium
• Eminla 0.071 minimum elastance of left
  atrium
• restVlas 13 systolic unloaded volume of
  left atrium
• restVlad 14 diastolic unloaded volume
  of left atrium
• Right atrium
• Emaxra 0.03 maximum elastance of the
  left atrium
• Eminra 0.027 minimum elastance of the
  left atrium

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Parameter values (Contd) and initial conditions

• Systemic circulation
  
• R1s 0.247 kPasec/ml Resistance of
  vessel 1
• R2s 0.051 kPasec/ml Resistance of
  vessel 2
• Rlv 7e-3 kPasec/ml Aortic impedance
• Rmit 5e-4 kPasec/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 kPasec/ml Resistance of
  vessel 1
• R2p 0.005 kPasec/ml Resistance of
  vessel 2
• Rrv 2e-3 kPasec/ml Arterial impedance
  of pulmonary artery
• Rtric 5e-4 kPasec/ml Resistance of
  tricuspid valve
• C1p 41.7 ml/kPa Compliance of
  vessel 1
• C2p 50 ml/kPa Compliance of vessel
  2
• Initial conditions
  

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Design time-varying elastances with modified
cosines(ventricular and atrial not drawn to
scale!)
systole
basic cycle length
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Make sure that unloaded volumes for LA, RA, LV
and RV vary accordingly between diastole and
systole
2
0
VR
P(t) E(t)V(t) - V0(t)
ESP
1
6
1
2
Pressure (kPa)
8

EDPVR
restVlvd
4

0

0
200

150

100

50
LV Volume (ml)
restVlvs
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More specifically (1)

• Solve the model from t0 to t3 sec in Matlab
  with ode23
• Use subplot to plot the following as a function
  of time 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 ejection fraction
  for the last beat?

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More specifically (2)

• Next, re-run the model, but at t3 sec well
  induce an acute left ventricular infarct.
  Simulate this by reducing LV maximum elastance by
  30.
• Stop the simulation at t10 sec.
• In figure 3, plot the pressure-volume diagram for
  the LV (for all beats from t0 to t10 sec).
• What is the stroke volume and ejection fraction
  for the last beat?
• Discuss what you observe.
• Remember to label your plots appropriately

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Hand in on paper (due date Thursday 9 April 2009)

• Matlab scripts (46 points)
• 3 figures (8 points each, total 24)
• Stroke volumes and ejection fractions for the
  normal and infarcted left ventricle (5 points
  each, total 20)
• Discussion for figure 3 (10 points)
• Also email your matlab scripts to Tyler
  (tseibert_at_ucsd.edu)