nonlinear systems term project: averaged modeling of the cardiovascular system

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?