Analysis of Medical Time Series Using Methods of Mathematical Physics Jan Kříž Department of...

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

YES !!!YES !!!

Is analysis of medical time series Is analysis of medical time series a suitable topic for M a suitable topic for M33Q school-conference?Q school-conference?

MM33Q: Q: Mathematical Methods in Quantum Mechanics

We exploit mathematical methods commonly used in quantum mechanics for data processing, namely:

• Differential geometry: Differential geometry: quantum waveguides theory• Maximum likelihood estimation:Maximum likelihood estimation: quantum state

reconstruction• Random matrix theory: Random matrix theory: quantum billiards

MOTIVATION

Why do we do this?Why do we do this?

MOTIVATION

Why do we do this?Why do we do this?

Quantum mechanics: no tradition in HK

Medical research has been provided inHK for more than fifty years.

Differential geometry & human cardiovascular dynamics measured by force

plate

Force plateForce plate

Measured are the three force and three momentum components, i.e. 6-dimensional multivariate time series


plate


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.)

Multivariate signal – processprocess: 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.


plate


plate

Curvatures - Curvatures - Geometrical invariants of a curveGeometrical 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 frameFrenet frame is a moving reference frame of orthonormal vectors which are used to describe a

curve locally at each point.


plate

To see a “Frenet frame” animationclick here


plate

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.

Frenet – Serret formulaeFrenet – Serret formulae

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

The mean over cardiac cycles was taken.

P-wave(systola of atria)

Q -wave

R-wave

S-wave

T-wave(repolarization)

QRS complex(systola of ventricles)


plate


plate


plateQuestion of interpretationQuestion 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

• 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.

Curvatures play a crucial role in spectral properties of Curvatures play a crucial role in spectral properties of quantum waveguidesquantum waveguides

EEG = electroencephalographyEEG = electroencephalography measures electric potentials on the scalp

(generated by neuronal activity in the brain)

MLE & human multiepoch EEG

Evoked potentialsEvoked potentials= responses to the external stimulus (auditory,

visual, etc.) sensory and cognitive processing in the brain


Basic concept of MLE 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.


Baryshnikov, B.V., Van Veen, B.D. and Wakai R.T., IEEE Trans. Biomed. Eng. 51 ( 2004), p. 1981 – 1993.

Experiment: Experiment: even, odd numbers recognition63 – channel EEG device 100 epochs

Assumptions: Assumptions: response is the same across all epochs,noise is independent from trial to trial,it is temporally white, but spatially colouredit is normally distributed with zero mean


Experiment: Experiment:


N … spatial channels , T … time samples per epochJ … number of epochs ( N=63, T=666, J=100)

data for j-th epoch: Xj = S + Wj ... N x T matrix

Estimate of repeated signal S in the form

S=HS=HCCTT

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, spatiallModel is purely linear, spatially-y-temporalltemporallyy nonlocal nonlocal


Filtering and averaging1. Filter data (4th order Butterworth filter with passband 1-20 Hz)2. Average data over all epochs- local in both temoral and spatial dimension


Commonly used methodCommonly used method


Results: channels 57-60Results: channels 57-60


Results: channels 25-28Results: channels 25-28


ResultsResults


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.

MLE MLE & quantum state reconstruction& quantum state reconstruction

Analysis of Medical Time Series Using Methods of Mathematical Physics Jan Kříž Department of...

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