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Mechanical manifestation of human cardiovascular
dynamicsJ.Kříž, P.Šeba
Department of physics,University of Hradec Kraloveand
K.Martiník, J. ŠťásekFaculty of Medicine, Charles University
QC workshop
“Spectra, Algorithms and Data Analysis“
February 28, 2006
Program
1. What is a force plate?
2. How to study cardiovascular system using force plate?
3. Differential geometry – method of data analysis
4. Results
5. Cardiac cycle
6. Comparing results (cardiac catetherization)
7. Interpretation
8. Conclusions
Force plate
Measured are the three force and three moment components, i.e. a six dimensional multivariate time series
Force plate – typical signals
only five independent channelsMF
Usual choice: force components + COP
,z
y
F
Mx .
z
x
F
My
Force plate
Typical COP (120 s) – spaghetti diagram
Our equipment
Experiment
Using the force plate and a special bed we measured the force plate output and the ECG signal on 20 healthy
adults. In three cases we measured also the heart sounds. In such a way we obtained a 7 or 8 dimensional time series. The used sampling rate was 1000 Hz. The
measurements lasted 8 minutes.
Typical measured signals
Periodic-like pattern of signals
Typical COP (10 s)
For a reclining subject the motion of the internal masses within the body has a crucial effect. Measured ground reaction forces contain information on the blood mass
transient flow at each heartbeat and on the movement of the heart itself. (There are also other sources of the internal
mass motion that cannot be suppressed, like the stomach activity etc, but they are much slower and do not display a
periodic-like pattern.)
Hypothesis
Multivariate signal – processprocess: multidimensional time-parameterized curve.
Measured channels: projections of the curve to given axes.
Example: changing the position of an electrode within EEG measurement changes the measured voltage. The measured process remains unchanged.
Measured forces and moments (projections) depend on the position of the pacient on the bed and on the position of the heart inside the body.
Characterizing the curve: geometrical invariants.
Method od data analysis
c: [a,b]Rn … Cn([a,b]) – mapping, such that
Length of a curve
Curvatures:
].,[,0)(' battc
dttclb
a )('
Geometrical invariants of a curve
The main message of the differential geometry: It is more natural to describe local properties of the curve in terms of a local reference system than using a global one like the euclidean coordinates.
Frenet frame is a moving reference frame of n orthonormal vectors ei(t) which are used to describe a curve
locally at each point c(t).
Frenet frame
Assume that are linearly independent
)(,),(''),(' )1( tctctc n].,[ bat
To see a “Frenet frame” animationclick here
The Frenet Frame is the family of orthonormal vectors called Frenet vectors. They are
constructed from the derivates of c(t) using the Gram-Schmidt orthogonalization algorithm with
The real valued functions are called generalized curvatures and are defined as
]},[|)(),(),({ 21 batttt n eee
).()()()(
,1,2 ),( )(),()()( ,)(
)()(
,)('
)(')(
121
1
1
)()(
1
tttt
nktttctctt
tt
tc
tct
nn
i
k
ii
kkk
k
kk
eeee
eeee
ee
e
1,,1 ),( njtj
.)('
)(),(')(
1
tc
ttt
jj
j
ee
Geometrical invariants: curvatures
2 – dimensional curve
3 – dimensional curve
)('
)('
)('
1)(,
)('
)('
)('
1)(
1
22
2
11 tc
tc
tct
tc
tc
tct ee
31221
1)('
)(')('')(')('')()(
tc
tctctctctt
…curvature
…tangent, normal
binormal)( normal,)( tangent,)( 321 ttt eee
31)('
)('')(')()(
tc
tctctt
22)('')('
)('''),('')(')()(
tctc
tctctctt
…curvature
…torsion
The simplest cases
Relation between the local reference frame and its changes
Main theorem of curve theory
.,,, curvatures has and 1)('
that so , curve ldimensiona- ations) transformEucleidian to(up unique is
Then there ).,( and 2,,1for 0)( with and 1,,1for
continuous- with ),( someon defined ,,, functionsGiven
121
1j121
n
j
jnn
ctc
cn
batnjtnj
Cba
Curvatures are invariant under reparametrization and Eucleidian transformations!
Therefore they are geometric properties of the curve.
Frenet – Serret formulae
The 5 curvatures were evaluated from 6 force plate signals.
Starting point of the cardiac cycle: QRS complex of ECG. Length of the cycle: approximately 1000 ms
Averaging
The mean over cardiac cycles was taken. Length of the cycle: approximately 1000 ms
P-wave(systola of atria)
Q -wave
R-wave
S-wave
T-wave(repolarization)
QRS complex(systola of ventricles)
Results
The results are reproducible
The question of interpretetion
The curvature maxima correspond to sudden changes of the curve, i.e. to rapid changes in the direction of the motion
of internal masses within the body.
The curvature maxima are associated with significant mechanical events, e.g. rapid heart expand/contract
movements, opening/closure of the valves, arriving of the pulse wave to various aortic branchings,...
Total blood circulation:
Veins right atrium right ventricle pulmonary artery lungs pulmonary vein left atrium left ventricle aorta branching to
capillares veins
Cardiac cycle
Cardiac cycle
Pressures inside the Heart
Pressure wave propagation along aorta
Ejected blood propagets in the form of the pressure wave
Pressure wave propagation along aorta
On branching places of large arteries the pulse wave is scattered and the subsequent elastic recoil contribute to the
force changes measured by the plate. A similar recoil is expected also when the artery changes its direction (like for
instance in the aortic arch).
Aorta and major branchings
Aortic arch
Diaphragm
Coeliac artery
Mesentric artery
Renalarteries
Abdominalbifurcation
Iliac arteries
Cardiac Catheterization involves passing a catheter (= a thin flexible tube) from the groin or the arm into the heart
produces angiograms (x-ray images)
can measure pressures in the left ventricle and the aorta
Cardiac Catheterization
For comparism we measured three volunteers on the force plate in the same day as they were catheterized.
Cardiac Catheterization
Pressures inside the Heart
Pressures inside the Heart – catheterization measurement
ECG
Ventricularpressure
Aortic pressure(aortal valve)
AVO
AVC
Pressures inside the Heart – catheterization measurement
ECG
Ventricularpressure
Aortic pressure(abdominal bifurcation)
Pressures in aorta
Aortic arch
Aortic valve
Pressures in aorta
Renal arteries
Diaphragm
Pressures in aorta
Arteria femoralis
Abdominal bifurcation
What is it good for?
Measuring the pressure wave velocity in large arteries
Observing pathological reflections (recoils)
Testing the effect of medicaments on the aortal wall properties
Testing the pressure changes in abdominal aorta in pregnant women
etc. and all this fully noninvasively. Cooperation of the patient is not needed
Conclusions
Depends on the elasticity of the arterial wall and on the arterial pressure.
Pressure wave velocity
Pressure wave velocity