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Sándor J Kovács PhD MD Washington University, St. Louis

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Discovering (predicting) new cardiac physiology/function from cardiac imaging, mathematical modeling and first principles

Sándor J Kovács PhD MD

Washington University, St. Louis

UCLA/IPAM 2/6/06

Imaging and modeling from cardiac imaging, mathematical modeling and first principles

allows us to go beyond

correlation to…

causality!

UCLA/IPAM 2/6/06

Focus: How the Heart Works When it Fills from cardiac imaging, mathematical modeling and first principles

The physiologic process by which the heart fills has confused cardiologists, physiologists, biomedical engineers, medical students and graduate students for generations.

UCLA/IPAM 2/6/06

How the Heart Works When it Fills from cardiac imaging, mathematical modeling and first principles

Why does it matter?

The recent recognition that up to 50% of patients

admitted to hospitals with congestive heart failure

have ‘normal systolic function’ as reflected by

ejection fraction, has further emphasized the

need to more fully understand the physiology

of diastole.

UCLA/IPAM 2/6/06

How the Heart Works When it Fills from cardiac imaging, mathematical modeling and first principles

In an effort to quantitate diastolic function using

a number or an index, the filling process has been

characterized via correlations of selected features of

either fluid (blood) flow or tissue displacement or

motion to LV ejection fraction, end-diastolic pressure

and other observables or clinical correlates such as

exercise tolerance or mortality.

UCLA/IPAM 2/6/06

How the Heart Works When it Fills from cardiac imaging, mathematical modeling and first principles

What do we know?

Anatomy

UCLA/IPAM 2/6/06

How the Heart Works: from cardiac imaging, mathematical modeling and first principlesanatomy

Pericardial anatomy

UCLA/IPAM 2/6/06

How the Heart Works: from cardiac imaging, mathematical modeling and first principlesanatomy

Pericardial anatomy

UCLA/IPAM 2/6/06

How the Heart Works When it Fills from cardiac imaging, mathematical modeling and first principles

Anatomy and terminology

UCLA/IPAM 2/6/06

How the Heart Works: from cardiac imaging, mathematical modeling and first principlesanatomy

Pericardial anatomy

UCLA/IPAM 2/6/06

How the Heart Works When it Fills from cardiac imaging, mathematical modeling and first principles

What else do we know?

Physiology

UCLA/IPAM 2/6/06

Doppler echocardiography reveals physiology from cardiac imaging, mathematical modeling and first principles:

Method by which transmitral

Doppler flow velocity data

is acquired

UCLA/IPAM 2/6/06

Echocardiographically observed patterns of filling from cardiac imaging, mathematical modeling and first principles:

S2 = second heart sound,

IR = isovolumic relaxation,

AT = acceleration time,

DT= deceleration time.

(Note: velocity scales differ slightly among images)

Waveform features (Epeak, E/A, DT, …) are correlated

with clinical aspects.

UCLA/IPAM 2/6/06

Cardiac catheterization reveals physiology from cardiac imaging, mathematical modeling and first principles:

Simultaneous, high fidelity LAP, LVP and transmitral Doppler

in closed chest canine. Note reversal of sign of A-V pressure gradient

As flow accelerates

(LAP > LVP) and decelerates (LAP < LVP).

Isovolumic

Relaxation

Rapid

Filling

Atrial

Systole

Diastasis

UCLA/IPAM 2/6/06

Cardiac catheterization reveals physiology from cardiac imaging, mathematical modeling and first principles:

Simultaneous aortic root,

LV pressure and LV volume as a function of time for one cardiac cycle

as measured in the cardiac catheterization laboratory.

dP/dV<0 at MVO

UCLA/IPAM 2/6/06

Cardiac catheterization reveals physiology from cardiac imaging, mathematical modeling and first principles:

AO

AVC

IVR

AVO

MVO

MVC

LA

LV

diastasis

atrial systole

Doppler A-wave

rapid filling

Doppler E-wave

UCLA/IPAM 2/6/06

Mechanics of filling from cardiac imaging, mathematical modeling and first principles:

Ventricle fills in 2 phases:

1) Early, rapid-filling (dP/dV< 0)

2) Atrial filling (dP/dV > 0)

(Actually, diastole has 4 phases: isovolumic relaxation, early rapid filling, diastasis, atrial contraction)

UCLA/IPAM 2/6/06

Catheterization and echo -combined from cardiac imaging, mathematical modeling and first principles

Abnormal

Pseudo-

Restriction

Restriction

relaxation

normalization

(reversible)

(irreversible)

Normal

40

0

N-

Mean LAP

TAU

NYHA

I-II

II-III

III-IV

IV

Grade

I

II

III

IV

UCLA/IPAM 2/6/06

How the Heart Works When it Fills from cardiac imaging, mathematical modeling and first principles

Recall key physiologic fact:

At -(and for a while after) - MVO, the LV simultaneously decreases its pressure while increasing its volume!

UCLA/IPAM 2/6/06

How the Heart Works When it Fills from cardiac imaging, mathematical modeling and first principles

We must therefore conclude that:

The heart is a suction pump in early diastole!

UCLA/IPAM 2/6/06

To go from from cardiac imaging, mathematical modeling and first principlescorrelation to causality devise a

kinematic model of suction initiated filling:

Newton’s Law: md2x/dt2 + c dx/dt + kx = 0

Initial conditions: x(0) = xo stored elastic strain to power suction

v(0) = 0 no flow prior to valve opening

Recall SHO has 3 regimes of motion,underdampedc2-4mk<0, critically dampedc2=4mk, overdampedc2 - 4mk>0.

VALIDATION: Compare model-predicted velocity of oscillator

to velocity of blood entering the ventricle through mitral valve.

UCLA/IPAM 2/6/06

Model of suction initiated filling from cardiac imaging, mathematical modeling and first principles:

Block-diagram of operational steps

Result: 1) re-express all E-and A-waves in terms of parameters AND

2) compute physiologic indexes

UCLA/IPAM 2/6/06

Model of suction initiated filling from cardiac imaging, mathematical modeling and first principles: does it fit the data?

Examples of model’s ability to fit in-vivo Doppler data

UCLA/IPAM 2/6/06

Model prediction compared to actual data: from cardiac imaging, mathematical modeling and first principles

Observed patterns of mitral valve inflow and superimposed model fits

S2 = second heart sound,

IR = isovolumic relaxation,

AT = acceleration time,

DT= deceleration time.

(Note: velocity scales differ slightly among images)

UCLA/IPAM 2/6/06

Kinematic model of suction initiated filling compared from cardiac imaging, mathematical modeling and first principles

to non-linear, coupled PDE models of filling:

Comparison of the PDF (red),

Meisner (blue) and Thomas (green) models for a clinical Doppler image. Note that all three models reproduce the contour of the image with comparable accuracy, and that the three models’ predictions are

essentially indistinguishable

graphically from one another.

UCLA/IPAM 2/6/06

Kinematic model of suction initiated filling from cardiac imaging, mathematical modeling and first principles:

Indexes from model parameters:

Mechanical Physiologic

kxo Force in spring Maximum A-V pressure

k Spring constant Chamber stiffness

1/2kxo2Stored energy Stored elastic strain

xoSpring displacement Velocity-time integral of E-wave

c2-4mk Regime of motion Stiff vs. delayed relaxation

UCLA/IPAM 2/6/06

Kinematic model of suction initiated filling from cardiac imaging, mathematical modeling and first principles:

Predictions from kinematic modeling:

1) The spring is linear and it is bi-directional

2) Underdamped, critically damped, overdamped regimes

3) Existence of ‘load independentindex’ of filling

4) Equilibrium volume of LV is diastasis

5) Tissue oscillations

6) Resonance

UCLA/IPAM 2/6/06

Kinematic model of suction initiated filling from cardiac imaging, mathematical modeling and first principles:

Physiologic analog and prediction of model:

Q: What is the spring?

UCLA/IPAM 2/6/06

What is the ‘spring’? from cardiac imaging, mathematical modeling and first principles

How the experiment that shows that cells can push was done!

Titin Develops Restoring Force in

Rat Cardiac Myocytes

Michiel Helmes, Károly Trombitás, Henk Granzier

Circulation Research. 1996;79:619-626.

UCLA/IPAM 2/6/06

What is the ‘spring’? from cardiac imaging, mathematical modeling and first principles

Experimental data proving that titin acts as a linear, bi-directional spring

It is hinged between thick and thin filaments.

UCLA/IPAM 2/6/06

Model of suction initiated filling from cardiac imaging, mathematical modeling and first principles:

Model can be used to

fit and (?) explain

heretofore unexplained

mechanism of biphasic

E-waves.

Early portion is governed by

k dominance, (underdamped)

later portion is governed by

c dominance (overdamped).

UCLA/IPAM 2/6/06

Kinematic modeling of filling from cardiac imaging, mathematical modeling and first principles:

“When you solve one difficulty, other new difficulties arise. You then try to solve them. You can never solve all difficulties at once.” P.A.M. Dirac

UCLA/IPAM 2/6/06

Modeling how the heart works: from cardiac imaging, mathematical modeling and first principles

Recall a physiologic fact -

Although the heart is an oscillator:

It is possible to remain (essentially) motionless!

UCLA/IPAM 2/6/06

Modeling how the heart works: from cardiac imaging, mathematical modeling and first principles

Hence:

The four-chambered heart is a

constant- volume pump!

UCLA/IPAM 2/6/06

How the Heart Works from cardiac imaging, mathematical modeling and first principles:(constant volume)

- Constant-volume attribute of the four-chambered heart -
- Hamilton and Rompf -1932
Hamilton W, Rompf H. Movements of the Base of the Ventricle and the Relative Constancy of the Cardiac Volume. Am J Physiol. 1932;102:559-65.

- Hoffman and Ritman -1985
Hoffman EA, Ritman E. Invariant Total Heart Volume in the Intact Thorax. Am J Physiol. 1985;18:H883-H890. Also showed that Left heart and Right heart are very nearly constant volume!

- Bowman and Kovács - 2003
Bowman AW, Kovács SJ. Assessment and consequences of the constant-volume attribute of the four-chambered heart. American Journal of Physiology, Heart and Circulatory Physiology 285:H2027-H2033, 2003.

- Hamilton and Rompf -1932

UCLA/IPAM 2/6/06

How the Heart Works When it Fills : from cardiac imaging, mathematical modeling and first principles (constant volume)

Cardiac MRI Cine Loop

‘four-chamber view”

Note relative absence

of ‘radial’ or ‘longitudinal’

pericardial surface

displacement or motion

UCLA/IPAM 2/6/06

How the Heart Works When it Fills : from cardiac imaging, mathematical modeling and first principles (constant volume)

Cardiac MRI Cine Loop

‘LV outflow track view”

Note relative absence

of ‘radial’ or ‘longitudinal’

pericardial surface

displacement or motion

UCLA/IPAM 2/6/06

How the Heart Works When it Fills : from cardiac imaging, mathematical modeling and first principles (constant volume)

Cardiac MRI Cine Loop

‘short-axis view”

Note slight ‘radial’ motion

of pericardial surface

UCLA/IPAM 2/6/06

How the Heart Works:( from cardiac imaging, mathematical modeling and first principlesconstant volume)

Cardiac MRI Cine Loop

‘four-chamber view”

Normal, human

UCLA/IPAM 2/6/06

How the Heart Works:( from cardiac imaging, mathematical modeling and first principlesconstant volume)

Cardiac MRI Cine Loop

‘short-axis view”

UCLA/IPAM 2/6/06

How the Heart Works:( from cardiac imaging, mathematical modeling and first principlesconstant volume)

Plot of # of pixels vs. frame number for 4-chamber slice

Diastole

Systole

Area (in Pixels)

Frame #

UCLA/IPAM 2/6/06

Rat heart - note almost ‘constant-volume’ feature from cardiac imaging, mathematical modeling and first principles

UCLA/IPAM 2/6/06

How the Heart Works:( from cardiac imaging, mathematical modeling and first principlesconstant volume)

Plot of # of voxels vs. fraction R-R interval for 3-D data set

Voxels

Fraction of R-R Interval

UCLA/IPAM 2/6/06

How the Heart Works:( from cardiac imaging, mathematical modeling and first principlesconstant volume)

Constant-Volume Attribute of the Four-Chambered Heart

Via MRI - how are images analyzed? (with Bowman, Caruthers, Watkins)

Conclusion: In normal, healthy subjects, the total volume enclosed within the pericardial sack remains constant to within a few percent. The pericardial surface exhibits only slight radial displacement throughout the cardiac cycle most notably along its diaphragmatic aspect.

UCLA/IPAM 2/6/06

How the Heart Works:( from cardiac imaging, mathematical modeling and first principlesconstant volume)

Cine MRI loop of pericardium for one cardiac cycle

UCLA/IPAM 2/6/06

How the Heart Works:( from cardiac imaging, mathematical modeling and first principlesconstant volume)

Right heart vs. left heart (n=20)

UCLA/IPAM 2/6/06

How the Heart Works:( from cardiac imaging, mathematical modeling and first principlesconstant volume)

What are predictable consequences of a constant volume, four-chambered heart

as they pertain to diastole?

(In light of the previous slide showing that the volumes of left and right heart are also independently constant.)

UCLA/IPAM 2/6/06

How the Heart Works:( from cardiac imaging, mathematical modeling and first principlesconstant volume)

Consider the motion of the mitral valve plane relative to the fixed apex and base.

Caltech 3/10/05

How the Heart Works:( from cardiac imaging, mathematical modeling and first principlesconstant volume)

One dimensional analog of mitral valve plane motion

atriumventricle

UCLA/IPAM 2/6/06

How the Heart Works:( from cardiac imaging, mathematical modeling and first principlesconstant volume)

Normalized MVP displacement vs. cardiac cycle

Normalized displacement

Percentage of cardiac cycle

UCLA/IPAM 2/6/06

atrioventricular from cardiac imaging, mathematical modeling and first principles

cross-section =A cm

2

Mitral valve area MVA cm

2

atrium

Mitral valve plane

-in diastole

myocardium

Mitral valve plane

velocity - Vmvp

Mitral valve plane

-in systole

ventricle

Modeling how the heart works:(constant volume)Concept: Consider a

simplified 2-chamber

constant-volume geometry

Application:Derive

the mitral annular velocity

(E’) to Doppler E-wave

(filling velocity) relation

UCLA/IPAM 2/6/06

Modeling from cardiac imaging, mathematical modeling and first principles how the heart works:(constant volume)

Conservation of volume for the upper and lower portions of the cylinder also imply tissue volume is conserved. How does the idealized LV chamber appear as it fills?

UCLA/IPAM 2/6/06

Modeling from cardiac imaging, mathematical modeling and first principles how the heart works:(constant volume)

Conservation of volume for the upper and lower portions of the cylinder imply:

Amvp Vmvp = Amv VE

At every instant during early rapid filling (Doppler E-wave)!

Note: Amvp and Amv are constant!!

UCLA/IPAM 2/6/06

Modeling from cardiac imaging, mathematical modeling and first principles how the heart works:(constant volume)

Conservation of volume means:

Amvp Vmvp = Amv VE

At every instant during early rapid filling (Doppler E-wave)!

Note: time varying quantity = time varying quantity

Rewrite as:

Amvp /Amv =VE /Vmvp

constant = constant

UCLA/IPAM 2/6/06

Modeling from cardiac imaging, mathematical modeling and first principles how the heart works: validation

Transmitral Doppler

Is constancy of

Amvp /Amv =VE /Vmvp really true?

UCLA/IPAM 2/6/06

Modeling from cardiac imaging, mathematical modeling and first principles how the heart works: validation

Mitral valve annular velocity via DTI

Is constancy of

Amvp /Amv =VE /Vmvp really true?

UCLA/IPAM 2/6/06

Modeling from cardiac imaging, mathematical modeling and first principles how the heart works: validation

Re-express the E and MVP velocity

Contours in terms of equivalent

contours using

PDF model and MBIP

(Note:time scale for lower picture

is expanded, velocity is plotted inverted)

UCLA/IPAM 2/6/06

Modeling from cardiac imaging, mathematical modeling and first principles how the heart works: validation

Overlay of VE and Vmvp on

the same velocity vs..

time coordinate axes.

Q1: What is peculiar about this?

Q2: What does it mean?

UCLA/IPAM 2/6/06

Modeling from cardiac imaging, mathematical modeling and first principles how the heart works: validation

mitral valve plane velocity (Vmvp) to Doppler E-wave (VE) relation - normal hearts

UCLA/IPAM 2/6/06

Modeling from cardiac imaging, mathematical modeling and first principles how the heart works: validation

Mitral valve plane velocity (Vmvp) to Doppler E-wave (VE) relation - data for

enlarged hearts!

UCLA/IPAM 2/6/06

Modeling from cardiac imaging, mathematical modeling and first principles how the heart works: validation + prediction

VE /VmvpRatio vs. LVEDP

Q1: it appears reasonably linear-

(r = 0.9196)WHY??

ANSWER: (- Hooke’s Law )

A = LVEDP

VE/ VMVP = (/MVA) LVEDP

UCLA/IPAM 2/6/06

How the Heart Works from cardiac imaging, mathematical modeling and first principles:(constant volume)

Relationship of [VE]max/ [Vmvp]max = E/E’ to

left ventricular end-diastolic pressure during simultaneous catheterization and echocardiography.

The ‘constant volume pump’ model predicted linear relationship is well fit by the data.

Best linear fit is provided by

E/E’ = 0.1753LVEDP + 1.8949 with r = 0.9196.

UCLA/IPAM 2/6/06

How the Heart Works :( from cardiac imaging, mathematical modeling and first principlesconstant volume)

E/E’ tabulated for 24 normal

subjects and 3 subjects with clinical CHF.

The model predicted value of @ 4 for the E/E’ relationship for the normal group is well fit by the data showing E/E’ = 4.4 ± 1.15. Three subjects with known CHF have greater than normal E/E’ @ 10, in accordance with model prediction.

UCLA/IPAM 2/6/06

Modeling from cardiac imaging, mathematical modeling and first principles how the heart works: prediction of load independent index of filling

Initial conditions

- m = inertia
- c = damping

- k = spring constant
- xo = initial displacement of spring

UCLA/IPAM 2/6/06

[1] from cardiac imaging, mathematical modeling and first principles

[2]

kxo=M(cEpeak)+b

Peak AV gradient (kxo)

Peak viscous force (cEpeak)

PredictedLoad Independent IndexChanges in preload change the shape of the E-wave, and thus must cause changes in k, c, and xo

The equation of motion, however, is obeyed regardless of changes in preload:

Consider the equation of motion at time of the E wave peak, t = tpeak

While 2 is true of any SHO, we invoke physiology:

[3]

Which implies:

[4]

Thus the maximum initial driving force (kxo) to peak attained viscous force (cEpeak) relation is predicted to be linear and load independent.

UCLA/IPAM 2/6/06

Modeling from cardiac imaging, mathematical modeling and first principles how the heart works: validation

- 15 healthy subjects (ages 20-30) with no history of heart disease and on no prescribed medication
- Subjects were positioned at three predetermined angles on a tilt-table. Data was acquired after transient heart rate changes resolved.
- E- and A-waves were recorded from subjects in supine, 90° head-up and 90° head-down positions.

UCLA/IPAM 2/6/06

Modeling from cardiac imaging, mathematical modeling and first principles how the heart works: validation

Load independent index of filling from kinematic modeling

UCLA/IPAM 2/6/06

Modeling from cardiac imaging, mathematical modeling and first principles how the heart works: validation

Load independent index of filling

UCLA/IPAM 2/6/06

Modeling from cardiac imaging, mathematical modeling and first principles how the heart works:prediction+validation

Load independent index of filling

Constant slope means

that the response to a

change in peak A-V

gradient is linear.

(data from all 15,

healthy volunteers)

UCLA/IPAM 2/6/06

High load filling regime from cardiac imaging, mathematical modeling and first principles

Supine load regime

Low load filling regime

Physiologic Interpretation of slopeHigh load

filling regime

Low load

filling regime

Low slope implies relatively larger increase in viscous loss for the same increase in peak driving force

Higher slope indicates greater efficiency in conversion of initial pressure gradient to attained filling volume.

Conclusions regarding load independent index from cardiac imaging, mathematical modeling and first principles

- Filling patterns change as load is altered, but changed filling patterns obey the same equation of motion. (F=ma)
- Proposed load independent index M obtainable from non-invasive Doppler Echo.
- Load independent index is defined by ratio of maximum driving force (kxo peak AV-gradient) to peak viscous force attained (cEpeak).
- The effect of pathology on M is unknown (so far) - but is predicted to be Mpathologic < M normal
- M is not expected to be uniquely associated with specific pathology, but will be different from normal.
- Greatest utility will be in comparing subjects to themselves in response to therapy

Summary conclusions: from cardiac imaging, mathematical modeling and first principles

- Unexplained correlations can be causallyexplained, and
- new cardiac physiology can be predicted from mathematical
- modeling and cardiac imaging.
- E-wave shapes predicted by SHO motion
- Bi-rectional, linear spring drives filling (TITIN)
- Constant-volume explains E’/E to LVEDP relation
- Load Independent index of filling, …

UCLA/IPAM 2/6/06

SEE desktop from cardiac imaging, mathematical modeling and first principles

Modeling, Imaging and Function from cardiac imaging, mathematical modeling and first principles

- Unsolved problems remain: (very incomplete listing)
- Relation between global and segmental indexes of filling
- What are the eigenvalues of diastolic function
- Can ‘optimal’ fillling function be defined
- Relation between model-parameters and biology
- Relation between model-parameters and pathology
- Relation between model-parameters and therapy
- Can you predict ‘stability’ vs ‘instability’ of oscillator?
- …
- …
- …

UCLA/IPAM 2/6/06

ACKNOWLEDGEMENTS: from cardiac imaging, mathematical modeling and first principles

NIH

AHA

VETERANS ADMINISTRATION

WHITAKER FOUNDATION

BARNES-JEWISH HOSPITAL FOUNDATION

ALAN A. AND EDITH L.WOLFF CHARITABLE TRUST

UCLA/IPAM 2/6/06

Modeling, Imaging and Function from cardiac imaging, mathematical modeling and first principles

THE END

UCLA/IPAM 2/6/06

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