Mechanical manifestationMechanical manifestation of human of human

hemodynamicshemodynamicsJ.Kříž, P.Šeba

Department of physics,University Hradec Kraloveand

K.MartiníkFaculty of Medicine, Charles University

15. konference českých a slovenských fyziků

7.9.2005

arXiv: physics/0507135

Force plate

Measured are the three force and three moment components , i.e. a six dimensional multivariate time series

Typical data

Force plate

Measured are the three force and three moment components , i.e. a six dimensional multivariate time series

only five independent channelsMF

Usual choice: three force components + point of application of the force: COP

,z

y

FM

x .z

x

FMy

Typical data: COP (120 s)

Our equipment

MeasurementsUsing the force plate and a special bed we measured the force plate output and the ECG signal on 17 healthy adult males. 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.

Typical data: COP (10 s)

For a reclining subject the motion of the internal masses withinthe 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-likepattern.)

Starting point of the cardiac cycle: the R wave of the ECG signal. Length of the cycle: 1000 ms

Multivariate signal: processprocess multidimensional time-parameterized curve.

Measured channel: projection of the curve to a given axis

Changing the position of an electrode within EEG measurement changes the measured voltage. The measured process remains unchanged.

Characterizing the curve: geometrical invariants:

c: [a,b] n … Cn([a,b]) – mapping, such that

examples of geometrical invariants: length of a curve

Curvatures

].,[,0)(' battc

dttclb

a )('

Frenet frameA Frenet frame is a moving reference frame of n orthonormal vectors e_i(t) which are used to describe a curve locally at each point γ(t).

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.

Assume that are linearly independent

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

)(,),(''),(' )1( tctctc n ].,[ bat

]},[|)(),(),({ 21 batttt n eee

).()()()(

,1,2 ),()(),()()( ,)()()(

,)(')(')(

121

1

1

)()(

1

tttt

nktttctctttt

tctct

nn

i

k

ii

kkk

k

kk

eeee

eeeeee

e

1,,1 ),( njtj

.)('

)(),(')( 1

tc

ttt jj

j

ee

Special cases2 – dimensional curve

3 – dimensional curve

)(')('

)('1)(,

)(')('

)('1)(

1

22

2

11 tc

tctc

ttctc

tct ee

31221

1)('

)(')('')(')('')()(tc

tctctctctt …curvature

…tangent, normal

binormal normal, tangent, )()()( 321 ttt eee

31)('

)('')(')()(

tc

tctctt

22)('')('

)('''),('')(')()(

tctc

tctctctt

…curvature

…torsion

Frenet-Serret FormulasRelation between the local reference frame and its changes

Main theorem of curve theory

.,,,1)('

).,(2,,10)(1,,1

),(,,,

121

1121

n

j

jnn

ctccn

batnjtnj

Cba

curvatures has and that so , curve ldimensiona- tions)transforma Eucleidian to (up unique is

there Then and for withand for

continuous- with some on defined functions Given j

Curvatures are invariant under reparametrization and Eucleidian transformations!Therefore they are geometric properties of the curve.

The 5 curvatures were evaluated at each cycle and the mean over cycles was taken. The measurement lasted 8 minutes

The results are reproducible

What does it mean?Are the curvature peaks linked to some physiological events?

On branching places of large arteries the pulse wave is scattered andthe 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 aortal arc).

Pressure wave oscillations

Pathology: abdominal aneurism

volunteer

pacient

Scattering of the pressure wave on the artery branchings / bendings leads to forces and moments measured by the force plate.

Pressure wave velocity :Depends on the elasticity of the arterial wall and on the arterial pressure.

Pulse wave velocity on large arteries is not directly accessible.

Timing and consistencyPulse wave velocity: c=L/T; L=0.7 m

Magnetic resonace measurements

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

etc. and all this fully noninvasively. Cooperation of the patient is not needed