nonlinear systems term project: averaged modeling of the cardiovascular system
TRANSCRIPT
AVERAGED MODELING OF THE CARDIOVASCULAR SYSTEMPhil Diette
SE 762
April 22, 2014
Introduction• Averaged Modeling of the Cardiovascular System
– Alexandru Codrean and Toma-Leonida Dragomir of Politehnica – IEEE Conference on Decision and Control
• Cycle-Averaged Dynamics of a Periodically Driven, Closed-Loop Circulation Model– Heldt, Chang, Chen, Verghese, Mark
• Papers analyzed for Etiometry– Data analytics for ICU
Outline• Pulsatile model of the cardiovascular system
• Problem Description• Characteristics of the Cardiovascular System• Electrical Circuit Analogy• Hybrid Linear System• Simulation
• Averaged model of the cardiovascular system• Averaging theory• Weighted average model• Simulation
Problem Description• Cardiovascular system - crucial physiological role• Mathematical modeling in medicine is a very active
research area• Wide range of models have been proposed• Detailed models can obscure basic functional principles• Simpler model needed to gain insight into dynamics of
physiological control• Stability of analyses of closed-loop cardiovascular control
• Baroreflex – negative feedback loop to control blood pressure
Characteristics of the Cardiovascular System
• Time-varying nonlinearities• Pulsatile flow: periodic variations• Heart valves: hybrid system with switching behavior
Characteristics of the Cardiovascular System
• Heart is a pump• Arteries and veins have elastance (compliance)• Filling a balloon
• Focused on left ventricle• Does not include
pulmonary circulation
Modeling
• Electrical circuit analogy
• Pulsatile lumped-parameters• Windkessel
ModelingCurrent = Flow
Electrical Resistance = Hydraulic Resistance
Diode = Heart Valve
Capacitance = Compliance
Voltage = Pressure
Charge = Volume
Time-varying Capacitance = Pulsatile Nature
Modeling
Heart
V0 = Cardiac Pressure
i1 = arterial blood flow
V2 = VenousPressure
V1 = Arterial Pressure
i2 = venous blood flow
i0 = cardiac blood flow
State-Space• Primary Equations:
State-Space• Primary Equations:
State-Space• Primary Equations:
• Heart pumping action
nT τ (n+1)T
Systole Diastole
Blood pushedout of heart
Blood flows backInto heart
State-Space • State Variables:
• Input:
• State Equations:
State-Space
• Rewrite system:
• Output:
Hybrid System• Recast the system as a hybrid system showing both
continuous and discrete dynamic behavior• Switching from systole to diastole regimes• Pulse frequency modulation• Rewrite input to reflect switching:
Hybrid System• Plug in switching u(t) and rewrite in the form of a
time-switched linear system
Hybrid System• Appropriate A and C matrices indexed by switching
function q(t)
Systole q(t)=0 Diastole q(t)=1
Reduced Order Model• Further simplification from 3rd order to a 2nd order model• Volume-charge analogy• Total stressed volume is known:
• Allows coupling with baroreflex
• Rewrite:
Reduced Order Model• 3rd order model to a 2nd order model
Simulation
Cs: 0.4 to 0.2 R1: 1 to 2 xT: 1034 to 1241
Averaging• Cardiovascular model has been simplified to linear
switched system, but complexity remains in its periodic and switched nature
• Two time scale problem• Baroreflex acts on a slower time scale by using time averaged
state variables• Cardiovascular system amenable to averaging techniques
• Many averaging techniques have been proposed, but these don’t suit simple analysis of cardiovascular closed-loop control
• Use weighted averages
Averaging Theory• Two time scale interpretation: if response of a system is
much slower than the input, then the response will be determined by the average of the input.
• Approximates solution by finding solution to an “average system”
• Stems from Perturbation Method• Classical perturbation method seeks an approximate solution as a
finite Taylor expansion of the exact solution• dx/dt = εf(t,x,ε), ε is a small positive parameter• Sufficiently small norm of ε, error will be small• Error is of order O(ε)
• These provide the technical basis for the averaging method
Averaging Theory• Khalil, Chapter 10• System:
• Autonomous average system:
Averaging Theory• Theorem:
Averaging Theory• Theorem, continued:
Averaging Theory• Theorem, continued:
Averaging Theory• Theorem, continued:
Weighted Averaging Method• State-space averaging• Apply weighted averaging operator, Mw, to periodic
function, ξ(t)
• a is a tuning parameter
Weighted Averaging Method• Apply weighted average to cardiovascular model:
Weighted Averaging Method• Rewrite:
θ is a function of T
Weighted Averaging Method• Offset error between trajectories of averaged system and
real averages of the original periodic system• Common for most averaging methods• Solution – use multiplicative tuning parameters
Simulation
Cs: 0.4 to 0.2 R1: 1 to 2 xT: 1034 to 1241
Summary• Modeled the cardiovascular system with lumped
parameters as an electrical circuit• Recast it as switched linear system• Simplified periodic system to an averaged autonomous
system with weighted averaging• Results
Questions?