Download Mechanical manifestation of human hemodynamics

Mechanical manifestation of human
hemodynamics
J.Kříž, P.Šeba
Department of physics,University Hradec Kralove
and
K.Martiník
Faculty of Medicine, Charles University
arXiv: physics/0507135
15. konference českých a slovenských fyziků
7.9.2005
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


FM
only five independent channels
Usual choice: three force components + point of application of the force: COP
x
My
Fz
,
Mx
y
.
Fz
Typical data: COP (120 s)
Our equipment
Measurements
Using 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
the body
of the internal masses within
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.)
Starting point of the cardiac cycle: the R wave of the ECG signal. Length of the
cycle: 1000 ms
Multivariate signal: process
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
c' (t )  0, t  [a, b].
examples of geometrical invariants:
length of a curve
b
l   c' (t ) dt
a
Curvatures
Frenet frame
A 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 c' (t ), c' ' (t ), , c
( n1)
(t ) are linearly independent t  [ a, b].
The Frenet Frame is the family of orthonormal vectors {e1 (t ), e 2 (t ), e n (t ) | t [a, b]}
called Frenet vectors. They are constructed from the derivates of c(t) using the
Gram-Schmidt orthogonalization algorithm with
c' (t )
e1 (t ) 
,
c' (t )
e k (t ) 
e k (t )
e k (t )
k 1
, e k (t )  c (t )   c ( k ) (t ), ei (t ) ei (t ), k  2,  n  1,
(k )
i 1
e n (t )  e1 (t )  e 2 (t )   e n 1 (t ).
The real valued functions  j (t ),
curvatures and are defined as
 j (t ) 
e' j (t ), e j 1 (t )
c' (t )
.
j  1,, n  1
are called generalized
Special cases
2 – dimensional curve
1
e1 (t ) 
c' (t )
c'1 (t ) 
c' (t ),
 2 
 (t )  1 (t ) 
1
e 2 (t ) 
c' (t )
 c'2 (t )
c' (t ) 
 1

 c' '1 (t )c'2 (t )  c' '2 (t )c'1 (t )
c' (t )
3
…curvature
3 – dimensional curve
e1 (t )tangent, e 2 (t )normal, e3 (t )binormal
 (t )  1 (t ) 
 (t )   2 (t ) 
c' (t )  c' ' (t )
c' (t )
3
…curvature
c' (t )  c' ' (t ), c' ' ' (t )
c' (t )  c' ' (t )
2
…torsion
…tangent, normal
Frenet-Serret Formulas
Relation between the local reference frame and its changes
Curvatures are invariant under reparametrization and Eucleidian transformations!
Therefore they are geometric properties of the curve.
Main theorem of curve theory
Given functions 1 ,  2 ,,  n 1 defined on some (a, b) with  j C n  j 1 - continuous
for j  1,, n  1 and with  j (t )  0 for j  1,, n  2 and t  (a, b). Then there
is unique (up to Eucleidian transforma tions) n - dimensiona l curve c, so that
c' (t )  1 and c has curvatures
1 ,  2 ,,  n 1.
The 5 curvatures were evaluated at each cycle and the mean
was taken. The measurement lasted 8 minutes
over cycles
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 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 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 consistency
Pulse 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