Detecting Targets in Human Body: What is the common property of radar systems and medical devices? Jan Kříž Department of physics, University of Hradec Králové Doppler Institute for mathematical physics and applied mathematics Joint work with Petr Šeba, Emil Doležal Tosa Yamada Sci-Tech Flash May 30, 2007 Kochi University of Technology

Program PART I 1.Introduction:What is the common property of radarsystems and medical devices? Which types of targets are we detecting in human body? 2.Motivation: Why do we do this? 3.Results: Caridovascular dynamics Processes in the brain 4.Conclusions:What is it good for?

Program PART II 1.Warming up: forces, moments and COP 2.Filtering 3.Differential geometry and force plate data analysis: curvatures as geometric invariants 4.Maximum likelihood estimation

RADAR = Radio Detection and Ranging

RADAR = Radio Detection and Ranging

EEG = Electroencephalography measures electric potentials on the scalp (generated by neuronal activity in the brain)

Multiepoch EEG: Evoked potentials = responses to the external stimulus (auditory, visual, etc.) sensory and cognitive processing in the brain

Force plate Measured are the three force and three momentum components (on strain-gauge technology). -stability analysis (balance in upright stance) -gait analysis

Human cardiovascular dynamics measured by force plate

ECG – electrocardiography measures electrical activity of the heart over time

Cardiac catheterizatrion  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

Cardiac Catheterization

What is the output? Summary

What is the common property of radar systems and medical devices? Output: m mm multivariate time series Signal processing: Signal processing: time series analysis Targets in human body: Targets in human body: processes in the brain, haemodynamical events, … spatial–temporalspatial–temporal character X = S + Wdata of the form X = S + W low signal to noise ratiolow signal to noise ratio (SNR)

MOTIVATION YES !!! Is this a suitable topic for a physicist? We exploit mathematical methods commonly used in quantum mechanics for data processing, namely: Differential geometry: Differential geometry: quantum waveguides theory general theory of relativity Maximum likelihood estimation: Maximum likelihood estimation:quantum state reconstruction Random matrix theory: Random matrix theory: quantum billiards Multivariate time series themselves are analyzed in physics: geophysics, climatology, meteorology, astrophysics,…

MOTIVATIONExample: JMA seismic intensity network Different types of rock layers filter the seismic waves. Aim of data analysis:source localization earthquake prediction

MOTIVATIONExample: Positions of electrodes Bones and coeliolymph filter the electric waves. Aim of data analysis:source localization seiuzure prediction

MOTIVATION Why do we do this?

Why do we do this? Quantum mechanics: no tradition in HK Medical research has been provided in HK for more than fifty years.

Force plate data analysis Typical signal measured during quiet standing

Postural requirements during quiet standing - support head and body against gravity - maintain COM within the base of support Force plate data analysis Postural control inputs Somatosensory systems (cutaneous receptors in soles of the feet, muscle spindle & Golgi tendon organ information, ankle joint receptors, proprioreceptors located at other body segments) Vestibular system (located in the inner ear) Visual system (the slowest one)

Force plate data analysis Typical COP (120 s) – spaghetti diagram

Motor strategies Motor strategies (to correct the sway) Force plate data analysis Ankle strategy (body = inverted pendulum, vertical forces) Hip strategy (larger and more rapid, shear forces) Stepping strategy

Postural control: Central nervous system (CNS) Force plate data analysis Spinal cord (reflex, 50 ms) Brainstem/subcortical (automatic response, 100 ms) Cortical (voluntary movements, 150 ms) Cerebellum -Our original goal: -Our original goal: study CNS using force plate data force plate as mechanical analog of EEG we have found some „strange“ latencies in the data.

Cardiovascular dynamics measured by force plate

Experiment Using the force plate and a special bed we measured the force plate output and the ECG signal on 20 healthy adults. In such a way we obtained a 7 dimensional time series. The used sampling rate was 1000 Hz. The measurements lasted 8 minutes. Cardiovascular dynamics measured by force plate

Typical measured signals 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.) Cardiovascular dynamics measured by force plate Ballistocardiography The idea is not new. Ballistocardiography (=usage of mikromovements for extracting information on the cardiac activity) is known for more than 70 years.

Cardiac cycle Cardiovascular dynamics measured by force plate Total blood circulation: Veins  right atrium  right ventricle  pulmonary artery  lungs  pulmonary vein  left atrium  left ventricle  aorta  branching to capillares  veins

Starting point of cycle: ventricle sys. ~ QRS of ECG. Length of the cycle: approximately 1000 ms The average over cardiac cycles is taken. P-wave (systola of atria) Q -wave R-wave S-wave T-wave (repolarization) QRS complex (systola of ventricles) Cardiovascular dynamics measured by force plate Mechanical activity is triggered by electric one.

Cardiovascular dynamics measured by force plate Data Filtering Averaging Black box (Curvatures)

Cardiovascular dynamics measured by force plate

Advantages of „Curvatures“ give more (and more precise) information than averaged forces / COP every curvature contains information on each measured channel do not depend on the position of the volunteer on the bed and on the position of the heart inside the body

Question of interpretation The curvature maxima correspond 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 assignment was done with the help of cardiac catheterization. Cardiovascular dynamics measured by force plate

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 Testing the pressure changes in abdominal aorta in pregnant women etc. and all this fully noninvasively. Cooperation of the patient is not needed Conclusions

Human multiepoch EEG „The analysis of EEG has a long history. Being used as a diagnostic tool for 70 years it still resists to be a subject of strict and objective analysis.“

Experiment: Human multiepoch EEG

Common property of evoked potentials and cardiovascular dynamics timelocked studied process is timelocked to some event. Cardiovascular dynamics is triggered by (QRS complex of) ECG signal. Evoked potentials are triggered by the instant of stimulus application. However, just described method does not work for evoked potentials.

Human multiepoch EEG low SNR The reason si: low SNR Noise – everything what we are not interested in, i.e. not only noise caused by imperfection of data acquisition – measured signal contains also other processes (not of interest) running inside the brain, resp. the body Cardiovascular dynamics: respiration, stomach activity… Evoked potentials: background activity of neurons Filtering + averaging: cardiovascular dynamics: OK evoked potentials: (sometimes still low SNR)

Human multiepoch EEG Data Filtering Averaging Black box (Curvatures) Data Black box 1 (MLE) Black box 2 (Curvatures, RMT)

Human multiepoch EEG – nonperiodic reversal Results: channels 57-60

Human multiepoch EEG – nonperiodic reversal Results: channels 25-28

Results Human multiepoch EEG – nonperiodic reversal

Results Human multiepoch EEG – needle sticking

Conclusions BETTER RESULTS THAN FILTERING/AVERAGING: low number of epochs low SNR

PART II Detecting Targets in Human Body: PART II Jan Kříž Department of physics, University of Hradec Králové Doppler Institute for mathematical physics and applied mathematics Joint work with Petr Šeba, Emil Doležal Tosa Yamada Sci-Tech Flash May 30, 2007 Kochi University of Technology

only five independent channels Usual choice: force components + COP Force plate

Filtering Generally, filtering is some mapping of a (univariate) time series: linear, nonlinear We need to filter out „unwanted“ frequencies: multiplying by a suitable function in the frequency domain.

process 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. 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. Curve:

Differential geometry & human cardiovascular dynamics measured by force plate Frenet frame 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. To see a “Frenet frame” animation click herehere

Differential geometry & human cardiovascular dynamics measured by force plate Assume that are lin. independent. The Frenet Frame is the family of orthonormal vectors called Frenet vectors. They are constructed from the derivates of c(t) using Gram-Schmidt orthogonalization, i. e.

Differential geometry & human cardiovascular dynamics measured by force plate The real valued functions are called generalized curvatures and are defined as Main theorem of curve theory

Differential geometry & human cardiovascular dynamics measured by force plate 2 – dimensional curve 3 – dimensional curve …curvature …tangent, normal …curvature …torsion

Differential geometry & human cardiovascular dynamics measured by force 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 formulae

Differential geometry & quantum waveguides theory Exner, Seba, J. Math. Phys. 30 (1989), Duclos, Exner, Rev. Math. Phys. 7 (1995), Krejcirik, JK, Publ. RIMS 41 (2005), Curvatures play a crucial role in spectral properties of quantum waveguides

Differential geometry & physics

Question of interpretation The curvature maxima correspond to the 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,... Cardiovascular dynamics measured by force plate

Pulse wave propagation Cardiovascular dynamics measured by force plate Ejected blood propagets in the form of the pressure wave

Pulse wave scattering Cardiovascular dynamics measured by force plate 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 aortic arch).

Aorta and major branchings Cardiovascular dynamics measured by force plate Aortic arch Diaphragm Coeliac artery Mesentric artery Renal arteries Abdominal bifurcation Iliac arteries

Assignment of curvature peaks to mechanical events: cardiac catheterization Cardiovascular dynamics measured by force plate For comparison we measured three volunteers on the force plate in the same day as they were catheterized.

Results Cardiovascular dynamics measured by force plate

Interpretation

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. MLE & human multiepoch EEG

Baryshnikov, B.V., Van Veen, B.D. and Wakai R.T., IEEE Trans. Biomed. Eng. 51 ( 2004), p – Assumptions: 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 MLE & human multiepoch EEG

N … spatial channels, T … time samples per epoch J … number of epochs data for j -th epoch: X j = S + W j... N x T matrix Estimate of repeated signal S in the form S=H q C T 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

Full dataset of J epochs: X=[ X 1 X 2... X J ]... N x JT matrix Noise over J epochs: W=[ W 1 W 2... W J ]...N x JT matrix X = [ S S... S ] + W, [ S S... S ] = H q D T, where D T = [ C T C T... C T ] Noise covariance „supermatrix“ is modeled as the Kronecker product of spatial and temporal covariance matrices, i.e. every element of N x N „spatial matrix“ is JT x JT „temporal matrix“ 1 R T = W T W… JT x JT temporal cov. matrix, ( R T =1 ) R = WW T … N x N spatial cov. matrix (unknown)

MLE & human multiepoch EEG Temporal basis matrix C Processes of interests in EEG are usually in the frequency band 1-20 Hz. Temporal basis vectors can be chosen as (discretized) sin(2 p ft), cos(2 p ft) to cover the frequency band of interest. The number of basis vectors L is given by frequency band. 1 In the case L=T we may choose C=1 (we take all frequencies)

MLE & human multiepoch EEG 1. Univariate normal distribution normally distributed random quantity x has pdf: where m is the mean and s 2 is the variance

MLE & human multiepoch EEG 2. Multivariate normal distribution Definition: The m x 1 random vector X is said to have m-variate normal distribution, if for every a  m the distribution of a T X is univariate normal. Mean:m=[ E(X 1 )... E(X m ) ] T Covariance matrix:S=E [( X – m)(X- m) T ] Theorem: If X is normally distributed then the pdf function is

MLE & human multiepoch EEG the pdf Under all above assumptions, the pdf can be written as 3. Normal distribution for multivariate time series Thus, we are looking for unknown matrices R, q and H to maximize the likelihood function for our data X. It was done by Baryshnikov et al.

MLE & human multiepoch EEG 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), , Springer, MLE & quantum state reconstruction