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
allows us to go beyond
correlation to…
causality!
UCLA/IPAM 2/6/06
Focus: How the Heart Works When it Fills
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
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
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
What do we know?
Anatomy
UCLA/IPAM 2/6/06
How the Heart Works:anatomy
Pericardial anatomy
UCLA/IPAM 2/6/06
How the Heart Works: anatomy
Pericardial anatomy
UCLA/IPAM 2/6/06
How the Heart Works When it Fills
Anatomy and terminology
UCLA/IPAM 2/6/06
How the Heart Works: anatomy
Pericardial anatomy
UCLA/IPAM 2/6/06
How the Heart Works When it Fills
What else do we know?
Physiology
UCLA/IPAM 2/6/06
Doppler echocardiography reveals physiology:
Method by which transmitral
Doppler flow velocity data
is acquired
UCLA/IPAM 2/6/06
Echocardiographically observed patterns of filling:
A
IR AT DT
S2
B
IR
S2
AT
DT
C
IR AT DT
S2
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:
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
Rapid
Relaxation
Filling
Diastasis
Atrial
Systole
UCLA/IPAM 2/6/06
Cardiac catheterization reveals physiology:
140
Simultaneous aortic root,
LV pressure and LV
volume as a function of
time for one cardiac cycle
as measured in the
cardiac catheterization
laboratory.
Pressures in mmHg and volume in ml
120
100
80
60
LVP
vol
Pao
40
dP/dV<0 at MVO
20
0
0
0.2
0.4
0.6
time t in seconds
0.8
1
1.2
UCLA/IPAM 2/6/06
Cardiac catheterization reveals physiology:
AO
IVR
AVC
AVO
MVO
MVC
LA
LV
rapid filling diastasis atrial systole
Doppler E-wave
Doppler A-wave
UCLA/IPAM 2/6/06
Mechanics of filling:
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
Normal
Abnormal
PseudoRestriction Restriction
relaxation normalization (reversible) (irreversible)
40
0
Mean LAP
N-
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
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
We must therefore conclude that:
The heart is a suction pump in
early diastole!
UCLA/IPAM 2/6/06
To go from correlation to causality devise a
kinematic model of suction initiated filling:
c
m
k
x(t), F(t)
Newton’s Law: m d2x/dt2 + c dx/dt + k x = 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, underdamped c2-4mk<0, critically
damped c2=4mk, overdamped c2 - 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:
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: does it fit the data?
Examples of model’s ability to fit in-vivo Doppler data
2.0
Velocity (m/s)
Velocity (m/s)
1.0
0.8
0.6
1.5
1.0
0.4
0.5
0.2
0.0
0.0
0.0
0.2
0.4
0.6
Time (s)
0.8
1.0
0.0
0.2
0.4
0.6
Time (s)
0.8
1.0
UCLA/IPAM 2/6/06
Model prediction compared to actual data:
Observed patterns of mitral valve inflow and superimposed model fits
A
IR AT DT
S2
B
IR
S2
AT
DT
C
IR AT DT
S2
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
to non-linear, coupled PDE models of filling:
1.8
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.
1.6
1.4
Velocity (m/s)
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Time (s)
UCLA/IPAM 2/6/06
Kinematic model of suction initiated filling:
Indexes from model parameters:
Mechanical
Physiologic
kxo
k
1/2kxo2
xo
c2-4mk
Force in spring
Spring constant
Stored energy
Spring displacement
Regime of motion
Maximum A-V pressure
Chamber stiffness
Stored elastic strain
Velocity-time integral of E-wave
Stiff vs. delayed relaxation
UCLA/IPAM 2/6/06
Kinematic model of suction initiated filling:
Predictions from kinematic modeling:
1) The spring is linear and it is bi-directional
2) Underdamped, critically damped, overdamped regimes
3) Existence of ‘load independent index’ 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:
Physiologic analog and prediction of model:
Q: What is the spring?
UCLA/IPAM 2/6/06
What is the ‘spring’?
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’?
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
(a)
Velocity (m/sec)
Velocity (m/sec)
Model of suction initiated filling:
0.8
0.6
0.4
0.2
0
0 0.2 0.4 0.6 0.8
Time (sec)
Model can be used to
fit and (?) explain
heretofore unexplained
mechanism of biphasic
E-waves.
0.8 E
A
0.6
0.4
Early portion is governed by
0.2
k dominance, (underdamped)
later portion is governed by
0
0 0.2 0.4 0.6 0.8 c dominance (overdamped). (
Time (sec)
UCLA/IPAM 2/6/06
Kinematic modeling of filling:
“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:
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:
Hence:
The four-chambered heart is a
constant- volume pump!
UCLA/IPAM 2/6/06
How the Heart Works :(constant volume)
•
Constant-volume attribute of the fourchambered 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:H883H890. 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 fourchambered heart. American Journal of 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 Loop
‘four-chamber view”
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
Note relative absence
of ‘radial’ or ‘longitudinal’
pericardial surface
displacement or motion
UCLA/IPAM 2/6/06
How the Heart Works When it Fills : (constant volume)
Cardiac MRI Cine Loop
‘LV outflow track view”
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
Note relative absence
of ‘radial’ or ‘longitudinal’
pericardial surface
displacement or motion
UCLA/IPAM 2/6/06
How the Heart Works When it Fills : (constant volume)
Cardiac MRI Cine Loop
‘short-axis view”
QuickTime™ and a
decompressor
are needed to see this picture.
Note slight ‘radial’ motion
of pericardial surface
UCLA/IPAM 2/6/06
How the Heart Works:(constant volume)
Cardiac MRI Cine Loop
‘four-chamber view”
Normal, human
QuickTime™ and a
YUV420 codec decompressor
are needed to see this picture.
UCLA/IPAM 2/6/06
How the Heart Works:(constant volume)
Cardiac MRI Cine Loop
‘short-axis view”
QuickTime™ and a
YUV420 codec decompressor
are needed to see this picture.
UCLA/IPAM 2/6/06
How the Heart Works:(constant volume)
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
Plot of # of pixels vs. frame number for 4-chamber slice
Systole
6000
Diastole
Area (in Pixels)
5000
4000
3000
2000
1000
0
2 4
6
8 10 12 14 16 18 20
Frame #
UCLA/IPAM 2/6/06
Rat heart - note almost ‘constant-volume’ feature
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
UCLA/IPAM 2/6/06
How the Heart Works:(constant volume)
Plot of # of voxels vs. fraction R-R interval for 3-D data set
1000
Diastole
Systole
LV + LA
RV + RA
Ao + PA
Pericardium
Voxels
800
600
400
200
0
0.0
0.2
0.4
0.6
0.8
1.0
Fraction of R-R Interval
UCLA/IPAM 2/6/06
How the Heart Works:(constant 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:(constant volume)
Cine MRI loop of pericardium for one cardiac cycle
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
UCLA/IPAM 2/6/06
How the Heart Works:(constant volume)
Right heart vs. left heart (n=20)
Average 4 Chamber
Normalized Area
1.2
1
0.8
Left
Right
Pericardium
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
% of cardiac cycl e
Average Short Axis Ventricles
Normalized Area
1.2
1
0.8
Left
Right
P ericardium
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
% of cardi ac cycl e
UCLA/IPAM 2/6/06
How the Heart Works:(constant 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:(constant volume)
Consider the motion of the mitral valve
plane relative to the fixed apex and base.
Caltech 3/10/05
How the Heart Works:(constant volume)
One dimensional analog of mitral valve plane motion
QuickTime™ and a
decompressor
are needed to see this picture.
atrium
ventricle
UCLA/IPAM 2/6/06
How the Heart Works:(constant volume)
Normalized displacement
Normalized MVP displacement vs. cardiac cycle
1
.
1
(
n
=
1
0
)
0
.
8
0
.
5
0
.
2
0
.
1
0
.
0
0
.
2
0
.
4
0
.
6
0
.
8
1
.
0
Percentage of cardiac cycle
UCLA/IPAM 2/6/06
Modeling how the heart works
:(constant volume)
atrioventricular
2
cross-section =A cm
Concept: Consider a
simplified 2-chamber
constant-volume geometry
Mitral valve area
MVA cm 2
atrium
myocardium
Mitral valve plane
velocity - Vmvp
Mitral valve plane
-in systole
Application: Derive
the mitral annular velocity
(E’) to Doppler E-wave
(filling velocity) relation
Mitral valve plane
-in diastole
ventricle
UCLA/IPAM 2/6/06
Modeling 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?
QuickTime™ and a
decompressor
are needed to see this picture.
UCLA/IPAM 2/6/06
Modeling 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 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 how the heart works
: validation
Transmitral Doppler
Is constancy of
Amvp /Amv =VE /Vmvp
really true?
UCLA/IPAM 2/6/06
Modeling 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
are
G
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
Velocity (m/sec)
Velocity (m/sec)
Modeling how the heart works
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
VE
Re-express the E and MVP velocity
Contours in terms of equivalent
contours using
PDF model and MBIP
0
0.2 0.4 0.6 0.8
Time (sec)
0.3
0.2
VMVP
0.1
0
: validation
(Note:time scale for lower picture
is expanded, velocity is plotted
inverted)
0 0.1 0.2 0.3 0.4
Time (sec)
UCLA/IPAM 2/6/06
Modeling how the heart works
: validation
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
Velocity (m/sec)

0.7












VE


0.6









0.5








0.4






0.3 






 






V



0.2  
MVP














0.1 















0
0
0.125
0.25
Time (sec)
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 how the heart works
: validation
mitral valve plane velocity (Vmvp) to Doppler Ewave (VE) relation - normal hearts
UCLA/IPAM 2/6/06
Modeling 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 how the heart works: validation
+
prediction
VE /Vmvp Ratio 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 :(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 :(constant volume)
E/E’ tabulated for 24 normal
subjects and 3 subjects with clinical CHF.
10
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.
9
8
Number of Subjects
7
6
5
4
3
2
1
0
<1 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5 •
12
VE/ V MVP
UCLA/IPAM 2/6/06
Modeling how the heart works
: prediction of load
independent index of filling
QuickTime™ and a
decompressor
are needed to see this picture.
QuickTime™ and a
decompressor
are needed to see this picture.
Finertia  Fdamping  Felastic  0
d2x
dx
m 2  c  kx  0
dx
dt


m = inertia
c = damping

x(0)  x o
xÝ(0)  0
Initial conditions
k = spring constant
xo = initial displacement of spring
UCLA/IPAM 2/6/06
Predicted Load Independent Index
Changes 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:
d2x
dx
m 2 c
 kx(t)  0
dt
dt
[1]
Consider the equation of motion at time of the E wave peak, t = tpeak

cE peak  kx(t peak )  0
[2]
While 2 is true of any SHO, we invoke physiology:

Which implies:
kxo=M(cEpeak)+b
kx(t 0 )  kx(t peak )
[3]
kx0  M(cE peak )  b
[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 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 how the heart works
: validation
Load independent index of filling from kinematic modeling
UCLA/IPAM 2/6/06
Modeling how the heart works
: validation
Load independent index of filling
UCLA/IPAM 2/6/06
Modeling how the heart works:prediction+validation
Load independent index of filling
Kxo vs. Emax*c for all
Constant slope means
that the response to a
change in peak A-V
gradient is linear.
subjects
50
y = 1.3432x + 4.2937
45
R2 = 0.98
40
35
(data from all 15,
healthy volunteers)
kxo
30
head down
25
supine
20
head up
15
Linear (Kxo
vs. EmaxC)
10
5
0
0
5
10
15
20
Emax*c
25
30
35
UCLA/IPAM 2/6/06
Physiologic Interpretation of slope
30
High load
filling
regime
20
Supine load regime
10
Low load
filling
regime
0
0
10
20
Peak Resistive Force peak
(cE)
M aximum Driving Force vs Peak
Resistive Force
Max Driving Force (Peak
AV Gradient kx o)
Max Driving Force (Peak
AV Gradient kx o)
M aximum Driving Force vs Peak
Resistive Force
30
30
High load
filling regime
20
10
Low load
filling regime
0
0
10
20
Peak Resistive Force peak
(cE)
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.
30
Conclusions regarding load independent index






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 noninvasive 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:
Unexplained correlations can be causally explained, 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
Modeling, Imaging and Function
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:
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
THE END
UCLA/IPAM 2/6/06