Transcript Force plate measurements of a human hemodynamics

Analysis of Medical Time Series Using
Methods of Mathematical Physics
Jan Kříž
Department of physics,
University of Hradec Králové
Doppler Institute
for mathematical physics and applied mathematics
Joint work with Petr Šeba
M3Q, Bressanone
February 26, 2007
MOTIVATION
Is analysis of medical time series
a suitable topic for M3Q school-conference?
YES !!!
M3Q: Mathematical Methods in Quantum Mechanics
We exploit mathematical methods commonly used in
quantum mechanics for data processing, namely:
• Differential geometry: quantum waveguides theory
• Maximum likelihood estimation: quantum state
reconstruction
• Random matrix theory: quantum billiards
MOTIVATION
Why do we do this?
MOTIVATION
Why do we do this?
Quantum mechanics:
no tradition in HK
Medical research has been provided in
HK for more than fifty years.
Differential geometry & human cardiovascular
dynamics measured by force plate
Force plate
Measured are the three force and three momentum
components, i.e. 6-dimensional multivariate time series
Differential geometry & human cardiovascular
dynamics measured by force plate
Differential geometry & human cardiovascular
dynamics measured by force plate
Differential geometry & human cardiovascular
dynamics measured by force plate
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.)
Differential geometry & human cardiovascular
dynamics measured by force plate
Multivariate signal – process: multidimensional timeparameterized curve.
Measured channels: projections of the curve to
given axes.
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. The measured
process remains unchanged.
Characterizing the curve: geometrical invariants.
Differential geometry & human cardiovascular
dynamics measured by force plate
Curvatures - 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
orthonormal vectors which are used to describe a
curve locally at each point.
Differential geometry & human cardiovascular
dynamics measured by force plate
To see a “Frenet frame” animation
click here
Differential geometry & human cardiovascular
dynamics measured by force plate
Frenet – Serret formulae
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.
On the other hand, the curve is uniquely (up to Eucleidian
transformations) given by its curvatures.
Differential geometry & human cardiovascular
dynamics measured by force plate
5 curvatures were evaluated from 6 force plate signals.
Starting point of cardiac cycle: QRS complex of ECG.
Length of the cycle: approximately 1000 ms
P-wave
(systola of atria)
R-wave
T-wave
(repolarization)
Q -wave
S-wave
QRS complex
(systola of ventricles)
The mean over cardiac cycles was taken.
Differential geometry & human cardiovascular
dynamics measured by force plate
Differential geometry & human cardiovascular
dynamics measured by force plate
Question of interpretation
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,...
The hypothesis was “proven“ by comparison of
measurements using force plate and cardiac
catheterization.
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 left ventricle and aorta
Differential geometry & quantum waveguides
theory
Curvatures play a crucial role in spectral properties of
quantum waveguides
• Exner, Seba, J. Math. Phys. 30 (1989), 2574-2580.
• Duclos, Exner, Rev. Math. Phys. 7 (1995), 73-102.
• Krejcirik, JK, Publ. RIMS 41 (2005), 757-791.
MLE & human multiepoch EEG
EEG = electroencephalography
measures electric potentials on the scalp
(generated by neuronal activity in the brain)
MLE & human multiepoch EEG
Evoked potentials
= responses to the external stimulus (auditory,
visual, etc.)
sensory and cognitive processing in the brain
MLE & human multiepoch EEG
MLE & human multiepoch EEG
Basic concept of MLE (R.A. Fisher in 1920’s)
• assume pdf f of random vector y depending on a
parameter set w, i.e. f(y|w)
• it determines the probability of observing the data
vector y (in dependence on the parameters w)
• however, we are faced with inverse problem: we have
given data vector and we do not know parameters
• define likelihood function l by reversing the roles of
data and parameter vectors, i.e. l(w|y) = f(y|w).
• MLE maximizes l over all parameters w
• that is, given the observed data (and a model of
interest), find the pdf, that is most likely to produce the
given data.
MLE & human multiepoch EEG
Baryshnikov, B.V., Van Veen, B.D. and Wakai R.T.,
IEEE Trans. Biomed. Eng. 51 ( 2004), p. 1981 – 1993.
Assumptions:
response is the same across all
epochs,
noise is independent from trial to trial,
it is temporally white, but spatially
coloured
it is normally distributed with zero
mean
Experiment:
even, odd numbers recognition
63 – channel EEG device
100 epochs
MLE & human multiepoch EEG
Experiment:
MLE & human multiepoch EEG
N … spatial channels ,
J … number of epochs
data for j-th epoch:
T … time samples per epoch
( N=63, T=666, J=100)
Xj = S + Wj ... N x T matrix
Estimate of repeated signal S in the form
S=HqCT
C … known T x L matrix of temporal basis vectors,
known frequency band is used to construct C
H … unknown N x P matrix of spatial basis vectors
q… unknown P x L matrix of coefficients
Model is purely linear, spatially-temporally nonlocal
MLE & human multiepoch EEG
Commonly used method
Filtering and averaging
1. Filter data (4th order Butterworth filter with
passband 1-20 Hz)
2. Average data over all epochs
- local in both temoral and spatial dimension
MLE & human multiepoch EEG
Results: channels 57-60
MLE & human multiepoch EEG
Results: channels 25-28
MLE & human multiepoch EEG
Results
MLE & human multiepoch EEG
MLE & quantum state reconstruction
Hradil, Řeháček, Fiurášek, Ježek,
Maximum
Likelihood Methods in Quantum Mechanics, in
Quantum State Estimation, Lecture Notes in Physics
(ed. M.G.A. Paris, J. Rehacek), 59-112, Springer,
2004.