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