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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, 2007Kochi University of Technology
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Program
PART IPART I1. Introduction: What is the common property of
radar systems 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?
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Program
PART II 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
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RADAR = Radio Detection and Ranging
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RADAR = Radio Detection and Ranging
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RADAR = Radio Detection and Ranging
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EEG = Electroencephalographymeasures electric potentials on the scalp
(generated by neuronal activity in the brain)
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Multiepoch EEG: Evoked potentials= responses to the external stimulus (auditory, visual, etc.)
sensory and cognitive processing in the brain
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Force plate
Measured are the three force and three momentum components (on strain-gauge technology).
- stability analysis (balance in upright stance)
- gait analysis
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Human cardiovascular dynamics measured by force plate
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ECG – electrocardiography measures electrical activity of the heart over time
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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
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Cardiac Catheterization
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What is the output?What is the output?
Summary
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What is the common property of radar systems and medical devices?
Output: multivariate time seriesmultivariate 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• data of the form X = S + WX = S + W
• low signal to noise ratiolow signal to noise ratio (SNR)
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MOTIVATION
YES !!!YES !!!
Is this a suitable topic for a physicist?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,…
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MOTIVATIONExample:Example:
JMA seismic intensity network
Different types of rock layers filter the
seismic waves.
Aim of data analysis: • source localization• earthquake prediction
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MOTIVATIONExample:Example:
Positions of electrodes
Bones and coeliolymph filter
the electric waves.
Aim of data analysis: • source localization• seiuzure prediction
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MOTIVATION
Why do we do this?Why do we do this?
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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.
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Force plate data analysis
Typical signal measured during quiet standingTypical signal measured during quiet standing
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Postural Postural rrequirementsequirements during quiet standing during quiet standing
- support head and body against gravity
- maintain COM within the base of support
Force plate data analysis
Postural Postural control inputscontrol inputsSomatosensory 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)
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Force plate data analysis
Typical Typical COP (120 s) – spaghetti diagramCOP (120 s) – spaghetti diagram
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Motor strategiesMotor 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
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Postural control: Postural control: Central nervous system (CNS)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 dataforce plate as mechanical analog of EEGwe have found some „strange“ latencies in the data.
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Cardiovascular dynamics measured by force plate
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ExperimentExperiment
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
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Typical measured signalsTypical measured signals
Cardiovascular dynamics measured by force plate
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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
The idea is not new. Ballistocardiography Ballistocardiography (=usage ofmikromovements for extracting information on the cardiac activity) is known for more than 70 years.
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Cardiac cycleCardiac 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
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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.
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Cardiovascular dynamics measured by force plate
DataFiltering
Averaging Black box
(Curvatures)
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Cardiovascular dynamics measured by force plate
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Cardiovascular dynamics measured by force plate
Advantages of „Curvatures“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
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Question of interpretationQuestion 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
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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
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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.“
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Experiment: Experiment:
Human multiepoch EEG
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Human multiepoch EEG
Common property of evoked potentialsCommon property of evoked potentialsand cardiovascular dynamicsand cardiovascular dynamics
studied process is timelockedtimelocked 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.
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Human multiepoch EEGThe reason si: low SNRlow 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)
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Human multiepoch EEG
DataFiltering
Averaging Black box
(Curvatures)
DataBlack box 1
(MLE)
Black box 2(Curvatures,
RMT)
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Human multiepoch EEG – nonperiodic reversal
Results: channels 57-60Results: channels 57-60
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Human multiepoch EEG – nonperiodic reversal
Results: channels 25-28Results: channels 25-28
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ResultsResults
Human multiepoch EEG – nonperiodic reversal
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ResultsResults
Human multiepoch EEG – needle sticking
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Conclusions
BETTER RESULTS THAN FILTERING/AVERAGING:
• low number of epochs• low SNR
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Detecting Targets in Human Body:
PART IIPART 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, 2007Kochi University of Technology
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only five independent channelsMF
Usual choice: force components + COP
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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.
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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.
Differential geometry & human cardiovascular dynamics measured by force
plate
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Differential geometry & human cardiovascular dynamics measured by force
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.
Curve: ].,[,0)('
,:
battc
Cbac nn
that such mapping, ba,
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Differential geometry & human cardiovascular dynamics measured by force
plate
Frenet frameFrenet frame is a moving reference frame of n orthonormal vectors ei(t) which are used to
describe a curve locally at each point.
To see a “Frenet frame” animationclick here
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Differential geometry & human cardiovascular dynamics measured by force
plateAssume that are lin. independent. )(,),(''),(' )1( tctctc n
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.
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Differential geometry & human cardiovascular dynamics measured by force
plateThe real valued functions are called generalized curvatures and are defined as
1,,1 ),( njtj
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Differential geometry & human cardiovascular dynamics measured by force
plate2 – dimensional curve
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tctc
tctctctt
…curvature
…torsion
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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 formulaeFrenet – Serret formulae
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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
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Differential geometry & physics
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Question of interpretationQuestion 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
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Pulse wave propagationPulse wave propagation
Cardiovascular dynamics measured by force plate
Ejected blood propagets in the form of the pressure wave
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Pulse wave scatteringPulse 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).
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Aorta and major Aorta and major branchingsbranchings
Cardiovascular dynamics measured by force plate
Aortic arch
Diaphragm
Coeliac artery
Mesentric artery
Renalarteries
Abdominalbifurcation
Iliac arteries
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Assignment of curvature peaks to Assignment of curvature peaks to mechanical events:mechanical events: cardiac catheterization 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.
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ResultsResults
Cardiovascular dynamics measured by force plate
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InterpretationInterpretation
Cardiovascular dynamics measured by force plate
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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
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Baryshnikov, B.V., Van Veen, B.D. and Wakai R.T., IEEE Trans. Biomed. Eng. 51 ( 2004), p. 1981 – 1993.
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
MLE & human multiepoch EEG
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N … spatial channels , T … time samples per epochJ … number of epochs
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
MLE & human multiepoch EEG
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MLE & human multiepoch EEG
Full dataset of J epochs: X=[ X1 X2 ... XJ ] ... N x JT matrixNoise over J epochs: W=[ W1 W2 ... WJ ] ...N x JT matrix
X = [ S S ... S ] + W ,
[ S S ... S ] = HDT, where DT = [ CT CT... CT ]
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“
RT= WTW… JT x JT temporal cov. matrix, (RT=11)R = WWT … N x N spatial cov. matrix (unknown)
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MLE & human multiepoch EEG
Temporal basis matrix Temporal basis matrix CCProcesses of interests in EEG are usually in the frequency band 1-20 Hz.
Temporal basis vectors can be chosen as (discretized) sin(2ft), cos(2ft) to cover the frequency band of interest.
The number of basis vectors L is given by frequency band.
In the case L=T we may choose C=11 (we take all frequencies)
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MLE & human multiepoch EEG
1. 1. Univariate normal distributionUnivariate normal distributionnormally distributed random quantity x has pdf:
where is the mean and 2 is the variance
2
2
2
)(exp
2
1),|(
x
xf
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MLE & human multiepoch EEG
2. 2. Multivariate normal distributionMultivariate normal distribution
Definition: The m x 1 random vector X is said to have m-variate normal distribution, if for every m the distribution of TX is univariate normal.
Mean: (X1) ... E(Xm)
Covariance matrix: X – X)T]
Theorem: If X is normally distributed then the pdf function is
)()(
2
1exp
2det
1),|( 1T
2/
XXXf
m
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MLE & human multiepoch EEG
Under all above assumptions, the pdfthe pdf can be written as
TTT1
2/2/))((Tr
2
1exp
2)(det
1),,|( DHXDHXR
RHRXf NTJTJ
33. . NoNormal distributionrmal distribution for multivariate time series for multivariate time series
Thus, we are looking for unknown matrices R, and H to maximize the likelihood function for our data X.
TTT1
2/2/))((Tr
2
1exp
2)(det
1)|,,( DHXDHXR
RXHRl NTJTJ
It was done by Baryshnikov et al.It was done by Baryshnikov et al.
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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), 59-112, Springer, 2004.
MLE MLE & quantum state reconstruction& quantum state reconstruction