Sándor J Kovács PhD MD Washington University, St. Louis

Sándor J Kovács PhD MDSándor J Kovács PhD MD

Washington University, St. Washington University, St. LouisLouis

UCLA/IPAM 2/6/06

Discovering (predicting) new Discovering (predicting) new cardiac physiology/function cardiac physiology/function

from cardiac imaging, from cardiac imaging, mathematical modeling and mathematical modeling and

first principlesfirst principles

UCLA/IPAM 2/6/06

Imaging and modeling Imaging and modeling allows us to go beyondallows us to go beyond

correlation correlation to…to…

causality!causality!

Focus: How the Heart Works When it FillsFocus: How the Heart Works When it Fills

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

UCLA/IPAM 2/6/06

How the Heart Works When it FillsHow the Heart Works When it Fills

The recent recognition that up to 50% of patients The recent recognition that up to 50% of patients admitted to hospitals with congestive heart failure admitted to hospitals with congestive heart failure have ‘normal systolic function’ as reflected by have ‘normal systolic function’ as reflected by ejection fraction, has further emphasized the ejection fraction, has further emphasized the need to more fully understand the physiology need to more fully understand the physiology of diastole.of diastole.

UCLA/IPAM 2/6/06

Why does it matter?Why does it matter?


In an effort to quantitate diastolic function usingIn an effort to quantitate diastolic function usinga number or an index, the filling process has been a number or an index, the filling process has been characterized via characterized via correlationscorrelations of selected features of of selected features of either fluid (blood) flow or tissue displacement or either fluid (blood) flow or tissue displacement or motion to LV ejection fraction, end-diastolic pressure motion to LV ejection fraction, end-diastolic pressure and other observables or clinical correlates such as and other observables or clinical correlates such as exercise tolerance or mortality.exercise tolerance or mortality.

UCLA/IPAM 2/6/06

What do we know?What do we know?

AnatomyAnatomy

How the Heart Works When it Fills

UCLA/IPAM 2/6/06

How the Heart Works:anatomy

How the Heart Works:anatomy

Pericardial anatomyPericardial anatomy

UCLA/IPAM 2/6/06

How the Heart Works: anatomy



UCLA/IPAM 2/6/06

Anatomy and terminology


UCLA/IPAM 2/6/06




UCLA/IPAM 2/6/06

What else do we What else do we know?know?

Physiology Physiology

How the Heart Works When it Fills

UCLA/IPAM 2/6/06

Doppler echocardiography reveals physiologyDoppler echocardiography reveals physiology:

Method by which transmitralDoppler flow velocity datais acquired

UCLA/IPAM 2/6/06

S2 = second heart sound, IR = isovolumic relaxation, AT = acceleration time, DT= deceleration time. (Note: velocity scales differ slightly among images)

S2

IR AT DT

S2

IR AT

S2

IR ATDT DT

A CB

UCLA/IPAM 2/6/06

Echocardiographically observed patterns of fillingEchocardiographically observed patterns of filling:

Waveform features (EWaveform features (Epeakpeak, E/A, DT, …) are , E/A, DT, …) are correlatedcorrelated with clinical aspects.with clinical aspects.

Simultaneous, high Simultaneous, high fidelity LAP, LVP fidelity LAP, LVP and transmitral and transmitral DopplerDopplerin closed chest in closed chest canine. Note canine. Note reversal of sign reversal of sign of A-V pressure of A-V pressure gradientgradientAs flow As flow accelerates accelerates (LAP > LVP) and (LAP > LVP) and decelerates (LAP < decelerates (LAP < LVP).LVP).

IsovolumicRelaxation

RapidFilling

DiastasisAtrial

Systole

UCLA/IPAM 2/6/06

Cardiac catheterization reveals physiologyCardiac catheterization reveals physiology:

LVP

vol

Pao

0 0.2 0.4 0.6 0.8 1 1.2

0

20

40

60

80

100

120

140

time t in seconds

Pressures in mmHg and volume in ml

Simultaneous Simultaneous aortic root, aortic root, LV pressure and LV LV pressure and LV volume as a volume as a function of time function of time for one cardiac for one cardiac cyclecycleas measured in the as measured in the cardiac cardiac catheterization catheterization laboratory.laboratory.

dP/dV<0 at dP/dV<0 at MVOMVO

UCLA/IPAM 2/6/06


rapid fillingDoppler E-waveDoppler E-wave

diastasisatrial systoleDoppler A-waveDoppler A-wave

AVC

MVO

MVC

AVO

IVR

AO

LVLA


UCLA/IPAM 2/6/06

Ventricle fills in 2 phases:Ventricle fills in 2 phases:

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

2) Atrial filling (dP/dV > 0)2) Atrial filling (dP/dV > 0)

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

Mechanics of fillingMechanics of filling:

UCLA/IPAM 2/6/06

Mean LAP

TAU

NYHA I-II II-III III-IV IV

Grade I II III IV

40

0

NormalAbnormalrelaxation

Pseudo-normalization

Restriction(reversible)

Restriction(irreversible)

N-

UCLA/IPAM 2/6/06

Catheterization and echo -combined

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


UCLA/IPAM 2/6/06

We must therefore conclude We must therefore conclude that:that:The heart is a The heart is a suction pumpsuction pump in in early diastole!early diastole!


UCLA/IPAM 2/6/06

c m k

x(t), F(t)

Recall SHO has 3 regimes of motion,Recall SHO has 3 regimes of motion, underdampedunderdamped cc22--4mk<04mk<0, , critically dampedcritically damped cc22=4mk=4mk, , overdampedoverdamped cc2 2 - 4mk>0- 4mk>0..

To go from To go from correlationcorrelation to to causality causality devise adevise akinematic model of suction kinematic model of suction initiated fillinginitiated filling:

UCLA/IPAM 2/6/06

Newton’s Law:Newton’s Law: m m dd22x/dtx/dt22 + + c c dx/dt dx/dt + + kk x = 0x = 0Initial conditions: x(0) = xInitial conditions: x(0) = xo o stored elastic strain to power suctionstored elastic strain to power suction v(0) = 0 v(0) = 0 no flow prior to valve openingno flow prior to valve opening

VALIDATION:VALIDATION: Compare model-predicted velocity of oscillator Compare model-predicted velocity of oscillator to velocity of blood entering the ventricle through mitral valve.to velocity of blood entering the ventricle through mitral valve.

Block-diagram of operational stepsBlock-diagram of operational steps

Result:Result: 1) re-express 1) re-express all all E-and A-waves in terms of parameters ANDE-and A-waves in terms of parameters AND2) compute physiologic indexes2) compute physiologic indexes

Model of suction initiated fillingModel of suction initiated filling:

UCLA/IPAM 2/6/06

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Velocity (m/s)

Time (s)

0.0

0.5

1.0

1.5

2.0

0.0 0.2 0.4 0.6 0.8 1.0

Velocity (m/s)

Time (s)

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

Model of suction initiated fillingModel of suction initiated filling: does it fit the data?

UCLA/IPAM 2/6/06

S2 = second heart sound, IR = isovolumic relaxation, AT = acceleration time, DT= deceleration time. (Note: velocity scales differ slightly among images)

S2

IR AT DT

S2

IR AT

S2

IR ATDT DT

A CB

Observed patterns of mitral valve inflow and superimposed model fits

UCLA/IPAM 2/6/06

Model prediction compared to actual data:Model prediction compared to actual data:

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Velocity (m/s)

Time (s)

Comparison of the PDF Comparison of the PDF ((redred), ), Meisner (Meisner (blueblue) and Thomas ) and Thomas ((greengreen) models for a ) models for a clinical Doppler image. clinical Doppler image. Note that all three Note that all three models reproduce the models reproduce the contour of the image with contour of the image with comparable accuracy, and comparable accuracy, and that the three models’ that the three models’ predictions are predictions are essentially essentially indistinguishable indistinguishable graphically from one graphically from one another.another.

Kinematic model of suction initiated filling comparedKinematic model of suction initiated filling comparedto non-linear, coupled PDE models of fillingto non-linear, coupled PDE models of filling:

UCLA/IPAM 2/6/06

Indexes from model Indexes from model parameters:parameters:

MechanicalMechanical PhysiologicPhysiologic

kxkxoo Force in springForce in spring Maximum Maximum A-V pressureA-V pressure

kk Spring constantSpring constant Chamber stiffnessChamber stiffness

1/2kx1/2kxoo22 Stored energyStored energy Stored elastic strainStored elastic strain

xxoo Spring displacementSpring displacement Velocity-time Velocity-time integral of E-waveintegral of E-wave

cc22-4mk-4mk Regime of motionRegime of motion Stiff vs. delayed Stiff vs. delayed relaxationrelaxation

Kinematic model of suction initiated fillingKinematic model of suction initiated filling:

UCLA/IPAM 2/6/06

Predictions from kinematic Predictions from kinematic modeling:modeling:

1) The spring is 1) The spring is linearlinear and it is and it is bi-directionalbi-directional

2) Underdamped, critically damped, overdamped 2) Underdamped, critically damped, overdamped regimesregimes

3) Existence of ‘3) Existence of ‘load independentload independent indexindex’ of ’ of fillingfilling

4) Equilibrium volume of LV is diastasis 4) Equilibrium volume of LV is diastasis

5) Tissue oscillations5) Tissue oscillations

6) Resonance6) Resonance


UCLA/IPAM 2/6/06

Physiologic analog and prediction of Physiologic analog and prediction of model:model:

Q: What is the spring?Q: What is the spring?


UCLA/IPAM 2/6/06

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

What is the ‘spring’?What is the ‘spring’?

UCLA/IPAM 2/6/06

Titin Develops Restoring Force in Rat Cardiac MyocytesMichiel Helmes, Károly Trombitás, Henk Granzier Circulation Research. 1996;79:619-626.

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

It is It is hinged hinged between between thick and thick and thin thin filaments.filaments.

What is the ‘spring’?What is the ‘spring’?

UCLA/IPAM 2/6/06

Model of suction initiated fillingModel of suction initiated filling:

QuickTime™ and aGraphics decompressorare needed to see this picture.

(b)0

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8Time (sec)

EA

(a)

0

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8Time (sec)

Model can be used to Model can be used to fit and (?) explain fit and (?) explain heretofore heretofore unexplainedunexplainedmechanism of biphasic mechanism of biphasic E-waves. E-waves.

Early portion is Early portion is governed by governed by kk dominance, dominance, (underdamped)(underdamped)later portion is later portion is governed by governed by cc dominance dominance (overdamped).(overdamped).UCLA/IPAM 2/6/06

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

Kinematic modeling of fillingKinematic modeling of filling:

UCLA/IPAM 2/6/06

Recall a physiologic Recall a physiologic fact - fact -

Although the heart is an Although the heart is an oscillator: oscillator:

It is possible to remain It is possible to remain (essentially) motionless!(essentially) motionless!

Modeling how the heart works:Modeling how the heart works:

UCLA/IPAM 2/6/06

Hence:Hence:

The four-chambered heart The four-chambered heart is a is a

constant- volumeconstant- volume pump! pump!

Modeling how the heart works:Modeling how the heart works:

UCLA/IPAM 2/6/06

How the Heart Works :(constant volume)


• Constant-volume attribute of Constant-volume attribute of the four-chambered heart -the four-chambered heart -• Hamilton and Rompf -1932 Hamilton and Rompf -1932

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

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

• Bowman and Kovács - 2003Bowman and Kovács - 2003 Bowman AW, Kovács SJ. Assessment and consequences of the constant-Bowman AW, Kovács SJ. Assessment and consequences of the constant-

volume attribute of the four-chambered heart. American Journal of volume attribute of the four-chambered heart. American Journal of Physiology, Heart and Circulatory Physiology 285:H2027-H2033, 2003.Physiology, Heart and Circulatory Physiology 285:H2027-H2033, 2003.

UCLA/IPAM 2/6/06

How the Heart Works When it Fills : (constant volume)


Cardiac MRI Cine LoopCardiac MRI Cine Loop

‘ ‘four-chamber view”four-chamber view”

Note relative absence Note relative absence of ‘radial’ or ‘longitudinal’of ‘radial’ or ‘longitudinal’pericardial surface pericardial surface displacement or motiondisplacement or motion

UCLA/IPAM 2/6/06

QuickTime™ and aCinepak decompressor

are needed to see this picture.




‘ ‘LV outflow track view”LV outflow track view”

Note relative absence Note relative absence of ‘radial’ or ‘longitudinal’of ‘radial’ or ‘longitudinal’pericardial surface pericardial surface displacement or motiondisplacement or motion

UCLA/IPAM 2/6/06






‘ ‘short-axis view”short-axis view”

Note slight ‘radial’ motionNote slight ‘radial’ motionof pericardial surface of pericardial surface

UCLA/IPAM 2/6/06

QuickTime™ and a decompressor


How the Heart Works:(constant volume)How the Heart Works:(constant volume)


‘‘four-chamber view”four-chamber view”

QuickTime™ and aYUV420 codec decompressor


Normal, humanNormal, human

UCLA/IPAM 2/6/06



‘‘short-short-

axis view”axis view”

QuickTime™ and aYUV420 codec decompressor


UCLA/IPAM 2/6/06


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


0

1000

2000

3000

4000

5000

6000

0 2 4 6 8 10 12 14 16 18 20

Systole Diastole

Frame #

Area (in Pixels)

UCLA/IPAM 2/6/06

UCLA/IPAM 2/6/06



Rat heart - note almost ‘constant-Rat heart - note almost ‘constant-volume’ featurevolume’ feature


0.0 0.2 0.4 0.6 0.8 1.0

0

200

400

600

800

1000

LV + LA

RV + RA

Ao + PA

Pericardium

Systole Diastole

Fraction of R-R Interval

Voxel

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

UCLA/IPAM 2/6/06

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:(constant volume)

Cine MRI loop of pericardium for one cardiac cycle

UCLA/IPAM 2/6/06




Right heart vs. left heart (n=20)Right heart vs. left heart (n=20) Average 4 Chamber

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

% of cardiac cycle

Normalized Area

Left

Right

Pericardium

Average Short Axis Ventricles

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

% of cardiac cycle

Normalized Area

Left

Right

Pericardium

UCLA/IPAM 2/6/06


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

as they pertain to diastole?as they pertain to diastole?

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

UCLA/IPAM 2/6/06


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

Caltech 3/10/05Caltech 3/10/05


One dimensional analog of mitral valve plane motionOne dimensional analog of mitral valve plane motion

atriumatrium ventricleventricle

UCLA/IPAM 2/6/06



How the Heart Works:(constant volume)

Normalized MVP displacement vs. cardiac cycle

Percentage of cardiac cycle

0.0 0.2 0.4 0.6 0.8 1.0

-0.1

0.2

0.5

0.8

1.1 (n = 10)

Normalized displacement

UCLA/IPAM 2/6/06

Modeling how the heart works :(constant volume)


atrium

myocardium

ventricle

atrioventricularcross-section =A cm

Mitral valve area MVA cm

Mitral valve plane-in diastole

Mitral valve planevelocity - Vmvp

Mitral valve plane-in systole

2

2

UCLA/IPAM 2/6/06

Application:Application: DeriveDerivethe mitral annular velocity the mitral annular velocity (E’) to Doppler E-wave (E’) to Doppler E-wave (filling velocity) relation(filling velocity) relation

Concept: Concept: Consider aConsider asimplified 2-chamber simplified 2-chamber constant-volume geometryconstant-volume geometry



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

UCLA/IPAM 2/6/06





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

AAmvpmvp V Vmvpmvp = A = Amvmv V VEE

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

Note: Note: AAmvp and mvp and AAmv are constant!!mv are constant!!

UCLA/IPAM 2/6/06



Conservation of volume means:

AAmvpmvp V Vmvpmvp = A = Amvmv V VEE

At every instant during early rapid filling At every instant during early rapid filling (Doppler E-wave)!(Doppler E-wave)!Note: time varying Note: time varying quantity = time varying quantityquantity = time varying quantity

Rewrite as:Rewrite as:

AAmvpmvp /A /Amvmv = =VVEE /V /Vmvpmvp

constant = constant = constantconstant

UCLA/IPAM 2/6/06

Modeling how the heart works : validation


Is constancy of Amvp /Amv =VE /Vmvp really true?

Transmitral Doppler

UCLA/IPAM 2/6/06



Mitral valve annular velocity via DTI

Is constancy of Is constancy of Amvp /AmvAmvp /Amv = =VVEE /Vmvp/Vmvp really really true?true?

UCLA/IPAM 2/6/06



Re-express the E and MVP velocity Contours in terms of equivalent contours usingPDF model and MBIP

(Note:time scale for lower pictureis expanded, velocity is plotted inverted)


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8Time (sec)

VE

0

0.1

0.2

0.3

0 0.1 0.2 0.3 0.4Time (sec)

VMVP

UCLA/IPAM 2/6/06



Overlay of VE and Vmvp on the same velocity vs..time coordinate axes.

Q1Q1: What is peculiar about this?: What is peculiar about this?Q2Q2: What does it mean?: What does it mean?


♦♦

♦

♦

♦

♦

♦

♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦

♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦

♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦

♦♦♦♦0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.125 0.25 ( )Time sec

VE

VMVP

UCLA/IPAM 2/6/06



mitral valve plane velocity (Vmitral valve plane velocity (Vmvpmvp) ) to Doppler E-wave (Vto Doppler E-wave (VEE) relation - ) relation - normal heartsnormal hearts


••••••••••••••••••••••••••••

•

•

•

0

2

4

6

8

10

12

14

16

18

20

0 0.04 0.08 0.12 0.16Time (sec)

UCLA/IPAM 2/6/06



Mitral valve Mitral valve plane velocity plane velocity (V(Vmvpmvp) to Doppler ) to Doppler E-wave (VE-wave (VEE) ) relation - data relation - data for for enlarged hearts!enlarged hearts!


• • • •

• • • • • • • ••

••

0

5

10

15

20

25

30

35

40

0 0.02 0.04 0.06 0.08Time (sec)

UCLA/IPAM 2/6/06

Modeling how the heart works: validation +

prediction

Modeling how the heart works: validation +

prediction

VVEE /Vmvp /Vmvp Ratio vs. LVEDPRatio vs. LVEDP

Q1Q1: it appears : it appears reasonably linear-reasonably linear-

((r = 0.9196)r = 0.9196) WHY??WHY??

ANSWERANSWER: (- Hooke’s Law ): (- Hooke’s Law )A = LVEDP

VE/ VMVP = (/MVA) LVEDP


♦♦

♦

♦

♦

♦

♦♦

♦♦ ♦

♦♦♦♦

♦

♦

♦

♦♦

♦

♦

♦

♦♦

♦♦♦

♦

♦

0

2

4

6

8

10

0 5 10 15 20 25 30 35LVEDP (mm Hg)

UCLA/IPAM 2/6/06



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

UCLA/IPAM 2/6/06

How the Heart Works :(constant volume)How the Heart Works :(constant volume)

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


<1 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5 ≥120

1

2

3

4

5

6

7

8

9

10

VE MVP/ V

E/E’ tabulated for 24 normal E/E’ tabulated for 24 normal subjects and 3 subjects with clinical CHF.subjects and 3 subjects with clinical CHF.

UCLA/IPAM 2/6/06

Modeling how the heart works : prediction of load independent index of filling

Modeling how the heart works : prediction of load independent index of filling

€

md2x

dx 2+ c

dx

dt+ kx = 0

€

Finertia + Fdamping + Felastic = 0

€

x(0) = xo

˙ x (0) = 0

mm = inertia = inertia

cc = damping = dampingkk = spring constant = spring constant

xxoo = initial displacement of spring = initial displacement of spring

QuickTime™ and a decompressorare needed to see this picture.QuickTime™ and a

decompressorare needed to see this picture.

Initial conditionsInitial conditions

UCLA/IPAM 2/6/06

PredictedPredicted Load Load Independent IndexIndependent IndexPredictedPredicted Load Load Independent IndexIndependent Index

Changes in preload change the shape of the E-wave, and thus must cause Changes in preload change the shape of the E-wave, and thus must cause changes in changes in kk, , cc, and , and xxoo

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

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

€

md2x

dt 2+ c

dx

dt+ kx(t) = 0 [1][1]

€

cE peak + kx(t peak ) = 0 [2][2]

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

€

kx(t0)∝ kx(t peak ) [3[3]]

Which implies:Which implies: [4[4]]

Thus the maximum initial Thus the maximum initial driving forcedriving force ( (kxkxoo) to ) to peak attained peak attained viscous forceviscous force ( (cEcEpeakpeak) relation is ) relation is predicted to be linear and load independent.predicted to be linear and load independent.

€

kx0 = M(cE peak ) + b

Peak AV gradient (kx

Peak AV gradient (kx oo))

Peak viscous force (cEPeak viscous force (cEpeakpeak))

kxkxoo=M(cE=M(cEpeakpeak)+b)+b

UCLA/IPAM 2/6/06



15 healthy subjects (ages 20-30) with no history 15 healthy subjects (ages 20-30) with no history of heart disease and on no prescribed medicationof heart disease and on no prescribed medication

Subjects were positioned at three predetermined Subjects were positioned at three predetermined angles on a tilt-table. Data was acquired after angles on a tilt-table. Data was acquired after transient heart rate changes resolved.transient heart rate changes resolved.

E- and A-waves were recorded from subjects in E- and A-waves were recorded from subjects in supine, 90° head-up and 90° head-down positions.supine, 90° head-up and 90° head-down positions.

UCLA/IPAM 2/6/06



Load independent index of filling from kinematic modeling

UCLA/IPAM 2/6/06



Load independent index of filling

UCLA/IPAM 2/6/06

Modeling how the heart works:prediction+validationModeling how the heart works:prediction+validation

Kxo vs. Emax*c for all subjects

y = 1.3432x + 4.2937

R2 = 0.98

0

5

10

15

20

25

30

35

40

45

50

0 5 10 15 20 25 30 35Emax*c

kxohead down

supine

head up

Linear (Kxovs. EmaxC)

Load independent index of filling

Constant slope means Constant slope means that the response to athat the response to achange in peak A-V change in peak A-V gradient is linear.gradient is linear.

(data from all 15, (data from all 15, healthy volunteers)healthy volunteers)

UCLA/IPAM 2/6/06

Physiologic Physiologic Interpretation of slopeInterpretation of slopePhysiologic Physiologic Interpretation of slopeInterpretation of slope

Maximum Driving Force vs Peak Resistive Force

0

10

20

30

0 10 20 30Peak Resistive Force (cEpeak )

Max Driving Force (Peak AV Gradient kx

o)

Low load filling regime

High load

filling regime

Supine load regime

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

Maximum Driving Force vs Peak Resistive Force

0

10

20

30

0 10 20 30Peak Resistive Force (cEpeak )

Max Driving Force (Peak AV Gradient kx

o)

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

High load filling regime

Low load filling regime

Conclusions regarding load Conclusions regarding load independent indexindependent index

Filling patterns change as load is altered, but Filling patterns change as load is altered, but changed filling patterns obey the same equation of changed filling patterns obey the same equation of motion. (F=ma)motion. (F=ma)

Proposed load independent index M obtainable from non-Proposed load independent index M obtainable from non-invasive Doppler Echo.invasive Doppler Echo.

Load independent index is defined by ratio of maximum Load independent index is defined by ratio of maximum driving force (driving force (kxkxoo peak AV-gradient) to peak viscous peak AV-gradient) to peak viscous force attained (force attained (cEcEpeakpeak). ).

The effect of pathology on M is unknown (so far) - but The effect of pathology on M is unknown (so far) - but is predicted to be Mis predicted to be Mpathologicpathologic < M < M normalnormal

M is not expected to be uniquely associated with M is not expected to be uniquely associated with specific pathology, but will be different from normal.specific pathology, but will be different from normal.

Greatest utility will be in comparing subjects to Greatest utility will be in comparing subjects to themselves in response to therapythemselves in response to therapy

Summary conclusions:Summary conclusions:

UCLA/IPAM 2/6/06

Unexplained Unexplained correlationscorrelations can be can be causallycausally explainedexplained, and , and newnew cardiac physiology can be predicted from mathematical cardiac physiology can be predicted from mathematical modeling and cardiac imaging.modeling and cardiac imaging.

• E-wave shapes predicted by SHO motionE-wave shapes predicted by SHO motion• Bi-rectional, linear spring drives filling (TITIN)Bi-rectional, linear spring drives filling (TITIN)• Constant-volume explains E’/E to LVEDP relationConstant-volume explains E’/E to LVEDP relation• Load Independent index of filling, …Load Independent index of filling, …

SEE desktop

Modeling, Imaging and FunctionModeling, Imaging and Function

UCLA/IPAM 2/6/06

Unsolved problemsUnsolved problems remain: (very incomplete listing) remain: (very incomplete listing)

Relation between global and segmental indexes of fillingRelation between global and segmental indexes of fillingWhat are the eigenvalues of diastolic functionWhat are the eigenvalues of diastolic functionCan ‘optimal’ fillling function be definedCan ‘optimal’ fillling function be defined

Relation between model-parameters and biologyRelation between model-parameters and biologyRelation between model-parameters and pathologyRelation between model-parameters and pathologyRelation between model-parameters and therapyRelation between model-parameters and therapyCan you predict ‘stability’ vs ‘instability’ of oscillator?Can you predict ‘stability’ vs ‘instability’ of oscillator?………………

NIH NIH AHA AHA VETERANS ADMINISTRATIONVETERANS ADMINISTRATIONWHITAKER FOUNDATIONWHITAKER FOUNDATIONBARNES-JEWISH HOSPITAL FOUNDATIONBARNES-JEWISH HOSPITAL FOUNDATIONALAN A. AND EDITH L.WOLFF CHARITABLE TRUSTALAN A. AND EDITH L.WOLFF CHARITABLE TRUST

UCLA/IPAM 2/6/06

ACKNOWLEDGEMENTS:ACKNOWLEDGEMENTS:

Modeling, Imaging and FunctionModeling, Imaging and Function

THE ENDTHE END

UCLA/IPAM 2/6/06

Documents

Sándor J Kovács PhD MD Washington University, St. Louis