Analysis of Medical Time Series Using Methods of Mathematical Physics

1
Analysis of Medical Time Series Using Methods of
Mathematical Physics

• Jan Kríž

Department of physics, University of Hradec
Králové Doppler Institute for mathematical
physics and applied mathematics
Joint work with Petr Šeba M3Q, Bressanone Febru
ary 26, 2007
2
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

3
MOTIVATION
Why do we do this?
4
MOTIVATION
Why do we do this?
Quantum mechanics no tradition in HK
Medical research has been provided in HK for more
than fifty years.
5
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
6
Differential geometry human cardiovascular
dynamics measured by force plate
7
Differential geometry human cardiovascular
dynamics measured by force plate
8
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.)
9
Differential geometry human cardiovascular
dynamics measured by force plate
Multivariate signal process multidimensional
time-parameterized 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.
10
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.
11
Differential geometry human cardiovascular
dynamics measured by force plate
To see a Frenet frame animation click here
12
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.
13
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
R-wave
P-wave (systola of atria)
T-wave (repolarization)
Q -wave
S-wave
QRS complex (systola of ventricles)
The mean over cardiac cycles was taken.
14
Differential geometry human cardiovascular
dynamics measured by force plate
15
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.
16
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

17
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.

18
EEG electroencephalography measures electric
potentials on the scalp (generated by neuronal
activity in the brain)
MLE human multiepoch EEG
19
Evoked potentials
MLE human multiepoch EEG
responses to the external stimulus (auditory,
visual, etc.) sensory and cognitive processing
in the brain
20
MLE human multiepoch EEG
21
Basic concept of MLE (R.A. Fisher in 1920s)
MLE human multiepoch EEG

• assume pdf f of random vector y depending on a
  parameter set w, i.e. f(yw)
• 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(wy)
  f(yw).
• 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.

22
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
23
MLE human multiepoch EEG
Experiment
24
MLE human multiepoch EEG
N spatial channels , T time samples
per epoch J number of epochs ( N63, T666,
J100)

• data for j-th epoch Xj S Wj ... N x T matrix
• Estimate of repeated signal S in the form
• SHqCT
• 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
  
• unknown P x L matrix of coefficients
• Model is purely linear, spatially-temporally
  nonlocal

25
Commonly used method
MLE human multiepoch EEG
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
26
Results channels 57-60
MLE human multiepoch EEG
27
Results channels 25-28
MLE human multiepoch EEG
28
Results
MLE human multiepoch EEG
29
MLE human multiepoch EEG
MLE quantum state reconstruction
Hradil, Rehácek, 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.