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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.