Clinical cardiovascular identification with Clinical cardiovascular identification with limited data and fast forward simulationlimited data and fast forward simulation

6th IFAC SYMPOSIUM ON MODELLING AND CONTROL IN BIOMEDICAL SYSTEMS

C. E. Hann1, J. G. Chase1, G. M. Shaw2, S. Andreassen3, B. W. Smith3

1Department of Mechanical Engineering, University of Canterbury, Christchurch, New Zealand2 Department of Intensive Care Medicine, Christchurch Hospital, Christchurch, New Zealand3 Centre for Model-based Medical Decision Support, Aalborg University, Aalborg, Denmark

• Problem: Cardiac disturbances difficult to diagnose– Limited data

– Reflex actions

• Solution: Minimal Model + Patient-Specific Parameter ID– Interactions of simple models to create complex dynamics

– Primary parameters

– Identification must use common ICU measurements

– E.g. increased resistance in pulmonary artery pulmonary embolism, atherosclerotic heart disease

• However: Identification for diagnosis requires fast parameter ID– Must occur in “clinical real-time”

– Limits model and method complexity (e.g. parameter numbers, non-linearities, …)

Diagnosis and TreatmentDiagnosis and Treatment

Heart ModelHeart Model

D.E.’s and PV diagramD.E.’s and PV diagram

2

2232

2

1

1121

1

21

L

RQPPQ

L

RQPPQ

QQV

2

0

)375.0(80

)(02

)(

),1())(1()()(

t

VVdes

ete

ePteVVEteP

Fast Forward SimulationFast Forward Simulation

• Formulate in terms of Heaviside functions:

• No looking for sign changes and restarting DE solver

done automatically

• E.g. filling (P1>P2, P2 ):

(close on flow)

2

22322322

1

11211211

2211

)()5.0)()((

)()5.0)()((

)()(

L

QRPPQHPPHHQ

L

QRPPQHPPHHQ

QQHQQHV

0)(,1

0)(,0))((

tK

tKtKH

)(

1tan)(tan

1

2

1))(( 11

tKtKtKH

0,,0

0,,1)5.0)()((

112

112121

QPP

QPPQHPPHH

Fast Forward SimulationFast Forward Simulation

• Ventricular interaction solve every time step (slow)

Substitute:

• Linear in Vspt Analytical solution very fast

)1())(1()()(

)1())(1()()(

)1())(1()()(

)(,0,

)(,0,

)(,0,,

,0

,0

,0

sptrvrvf

sptlvlvf

sptsptspt

VVrvfsptrvrvfes

VVlvfsptlvlvfes

VVsptsptdsptsptes

ePteVVEte

ePteVVEte

ePteVVEte

rvfrvrvfV

lvflvlvfV

sptsptsptV

bVaebVaebVae rvrvflvlvfsptspt ,,

Method (20 Heart beats) CPU time (s) Speed increase factor

Event solver 101.9

First Heaviside 36.8 2.8

Simpler Heaviside 18.8 5.4

Simpler Heaviside + Vspt formula 3.1 32.9

Reflex ActionsReflex Actions• Vaso-constriction - contract veins

• Venous constriction – increase venous dead space

• Increased HR

• Increased ventricular contractility

Varying HR as a linear function of Pao

Simple interactions to create overall complex dynamic behaviour in the full system model

Disease StatesDisease States

• Pericardial Tamponade:– Build up of fluid in pericardium– Decrease: dead space volume V0,pcd

• Pulmonary Embolism:– Increase: Rpul

• Cardiogenic shock:– Not enough oxygen to myocardium (e.g. from blocked coronary artery)– Decrease: Ees,lvf, Increase: P0,lvf A more complex set of changes/interactions

• Septic shock:– Blood poisoning– Decrease: Rsys = systemic resistance

• Hypovolemic shock:– Severe decrease in total blood volume = sum of individual volumes

Current Status:• Clinical results for Pulmonary Embolism• Simulated results for others

A Healthy Human BaselineA Healthy Human Baseline

Output Value

Volume in left ventricle 111.7/45.7 ml

Volume in right ventricle 112.2/46.1 ml

Cardiac output 5.3 L/min

Max Plv 119.2 mmHg

Max Prv 26.2 mmHg

Pressure in aorta 116.6/79.1 mmHg

Pressure in pulmonary artery 25.7/7.8 mmHg

Avg pressure in pulmonary vein 2.0 mmHg

Avg pressure in vena cava 2.0 mmHg

Simulation of Disease StatesSimulation of Disease States

• Pericardial Tamponade Ppu – 7.9 mmHgCO – 4.1 L/minMAP – 88.0 mmHg

• Pulmonary Embolism: Rpul increases as embolism induced

• All other disease states similarly capture physiological trends and magnitudes

~60-75% change in mean value

Integral Method - ConceptIntegral Method - Concept

Discretised solution analogous to measured data

• Work backwards and find a,b,c

• Current method – solve D. E. numerically or analytically

8.0,2.0,5.0

1)0(,)sin(

cba

xctbaxx• (simple example with analytical solution )

R

PPq 21

12 13 14 15 16 17 18 191.4

1.45

1.5

1.55

1.6

1.65

1.7

1.75

1.8

1.85

time

x

))sincos(

)(()1(

1)(

22

322

ccatbatab

acaabcaeaa

tx at

- Find best least squares fit of x(t) to the data

- Non-linear, non-convex optimization, computationally intense

• integral method

– reformulate in terms of integrals

– linear, convex optimization, minimal computation

Integral Method - ConceptIntegral Method - Concept

0t• Integrate both sides from to t ( ) ,)sin( ctbaxx 4

0t

t

t

t

t

t

t

t

t

t

t

t

t

ttcttbdtxatxtx

dtcdttbdtxatxtx

dtctbaxdtx

0

0 0 0

00

)())cos()(cos()()(

1)sin()()(

))sin((

000

0

• Choose 10 values of t, between and form 10 equations in 3 unknowns a,b,c

40t 6

10,,1),()()())cos(1( 000

itxtxttctbdtxa iitt i

Integral Method - ConceptIntegral Method - Concept

)()(

)()(

)cos()cos(

)cos()cos(

010

01

010100

0110

10

0

1

0

txtx

txtx

c

b

a

ttttdtx

ttttdtx

tt

tt

• Linear least squares (unique solution)

Method Starting point CPU time (seconds) Solution

Integral - 0.003 [-0.5002, -0.2000, 0.8003]

Non-linear [-1, 1, 1] 4.6 [-0.52, -0.20, 0.83]

Non-linear [1, 1, 1] 20.8 [0.75, 0.32, -0.91]

• Integral method is at least 1000-10,000 times faster depending on starting point

• Thus very suitable for clinical application

Adding Noise & IDAdding Noise & ID

• Add 10% uniform distributed noise to outputs to identify

• Conservative value that is larger than what has been encountered clinically

• Sample at 100-200 Hz

• Capture disease states, assume Ppa, Pao, Vlv_max, Vlv_min, chamber flows.

Identifying Disease State Using Identifying Disease State Using All Variables - SimulatedAll Variables - Simulated

Change True value

(ml)

Optimized Value

Error (%)

First 180 176 2.22

Second 160 158 1.25

Third 140 138 1.43

Fourth 120 117 2.50

Fifth 100 100 0

• Pericardial Tamponade: Find all parameters, look for change in V0,pcd

• Pulmonary Embolism: Find all parameters, look for change in Rpul

Change True value

(mmHg s ml-1)

Optimized Value Error (%)

First 0.1862 0.1907 2.41

Second 0.2173 0.2050 5.67

Third 0.2483 0.2694 8.50

Fourth 0.2794 0.2721 2.60

Fifth 0.3104 0.2962 4.59

• All other variables effectively unchanged from baseline

• All other variables effectively unchanged from baseline

• Cardiogenic shock: Find all parameters, look for change in [Ees,lvf, P0,lvf]

Change True values Optimized Value

Error (%)

First [2.59,0.16] [2.61,0.15] [0.89,5.49]

Second [2.30,0.19] [2.30,0.18] [0.34,4.39]

Third [2.02,0.23] [2.02,0.21] [0.43,8.03]

Fourth [1.73,0.26] [1.70,0.24] [1.48,9.85]

Fifth [1.44,0.30] [1.43,0.27] [0.47,9.39]

Change True value

(mmHg s ml-1)

Optimized Value

Error (%)

First 1.0236 1.0278 0.41

Second 0.9582 0.9714 1.37

Third 0.8929 0.8596 3.73

Fourth 0.8276 0.8316 0.49

Fifth 0.7622 0.7993 4.86

• Septic Shock: Find all parameters, look for change in Rsys

Identifying Disease State Using Identifying Disease State Using All Variables - SimulatedAll Variables - Simulated

• All other variables effectively unchanged from baseline

• All other variables effectively unchanged from baseline

• Hypovolemic Shock: Find all parameters, look for change in total stressed blood volume

Change True value

(ml)

Optimized Value

Error (%)

First 1299.9 1206.5 7.18

Second 1177.3 1103.7 6.26

Third 1063.1 953.8 10.28 (hmm!)

Fourth 967.8 1018.9 5.28

Fifth 928.5 853.4 8.10

Identifying Disease State using Identifying Disease State using All Variables - SimulatedAll Variables - Simulated

• All other variables effectively unchanged from baseline

New for MCBMSNew for MCBMS! Clinical Results! Clinical Results

• Pulmonary embolism induced in pigs (collaborators in Liege, Belgium)

• Identify changes in Rpul including reflex actions and all variables

Left Ventricle (30 mins) Right Ventricle (30 mins)

• Use only: Pao, Ppa, min/max(Vlv, Vrv) to ID all parameters

• Far fewer measurements than in simulation

Animal Model Results – PV loopsAnimal Model Results – PV loops

Left Ventricle (120 mins)

Right Ventricle (120 mins)

Left Ventricle (180 mins)

Right Ventricle (120 mins)

Animal Model Results Over TimeAnimal Model Results Over Time

SeptumVolume(Pig 2)

Right ventricleexpansion index(Pig 2)

Pulmonary resistance(Pig 2)

Reflex actions(Pig 2)

PV loops (Pig 2)

Rpul – All pigs

ConclusionsConclusions

• Minimal cardiac model simulate time varying disease states– Accurately captures physiological trends and magnitudes

– Accurately captures a wide range of dynamics

– Fast simulation methods available

• Integral-based parameter ID patient specific models– Simulation: ID errors from 0-10%, with 10% noise

– Animal model: Pressures (Total Error) =2.22±2.17 mmHg (< 5%)

Volumes (Total Error) = 2.37±2.01 ml (< 5%)

• Identifiable using a minimal number of common measurements– Rapid ID method, easily implemented in Matlab

– Rapid ID = Rapid diagnostic feedback

• Future Work = septic shock (Dec 2006), and other disease states (2007)

AcknowledgementsAcknowledgements

Questions ???

Engineers and Docs

Dr Geoff Chase Dr Geoff Shaw

The Danes

Prof Steen Andreassen Dr Bram Smith

The Honorary Danes

Belgium Honorary Kiwis

Dr Thomas Desaive

Christina Starfinger