detecting targets in human body: what is the common property of radar systems and medical devices?...

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

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?

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

RADAR = Radio Detection and Ranging

EEG = Electroencephalographymeasures 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?What is the output?

Summary

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)

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,…

MOTIVATIONExample:Example:

JMA seismic intensity network

Different types of rock layers filter the

seismic waves.

Aim of data analysis: • source localization• earthquake prediction

MOTIVATIONExample:Example:

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?

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.

Force plate data analysis

Typical signal measured during quiet standingTypical signal measured during quiet standing

Postural Postural rrequirementsequirements during quiet standing during quiet standing

- support head and body against gravity

- maintain COM within the base of support


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)


Typical Typical COP (120 s) – spaghetti diagramCOP (120 s) – spaghetti diagram

Motor strategiesMotor strategies (to correct the sway)


Ankle strategy (body = inverted pendulum, vertical forces)

Hip strategy (larger and more rapid, shear forces)

Stepping strategy

Postural control: Postural control: Central nervous system (CNS)Central nervous system (CNS)


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.

Cardiovascular dynamics measured by force plate

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.


Typical measured signalsTypical measured signals


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.)


The idea is not new. Ballistocardiography Ballistocardiography (=usage ofmikromovements for extracting information on the cardiac activity) is known for more than 70 years.

Cardiac cycleCardiac cycle


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)


Mechanical activity is triggered by electric one.


DataFiltering

Averaging Black box

(Curvatures)


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

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.


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: Experiment:



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.

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)


DataFiltering

Averaging Black box

(Curvatures)

DataBlack box 1

(MLE)

Black box 2(Curvatures,

RMT)

Human multiepoch EEG – nonperiodic reversal

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


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

ResultsResults


ResultsResults

Human multiepoch EEG – needle sticking

Conclusions

BETTER RESULTS THAN FILTERING/AVERAGING:

• low number of epochs• low SNR

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

only five independent channelsMF

Usual choice: force components + COP

,z

y

F

Mx .

z

x

F

My

Force plate

FrM

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.

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


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,


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


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.

]},[|)(),(),({ 21 batttt n eee

).()()()(

,1,2 ),( )(),()()( ,)(

)()(

,)('

)(')(

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1

tttt

nktttctctt

tt

tc

tct

nn

i

k

ii

kkk

k

kk

eeee

eeee

ee

e


plateThe real valued functions are called generalized curvatures and are defined as

1,,1 ),( njtj

.)('

)(),(')(

1

tc

ttt

jj

j

ee

Main theorem of curve theory

.,,,1)('

).,(2,,1

0)(1,,1

),(,,,

121

1

121

n

jjn

n

ctc

cn

batnj

tnjC

ba

curvatures has and that so

, curve ldimensiona- tions)transforma Eucleidian to (up

unique is there Then and for

withand for continuous-

with some on defined functions Given j


plate2 – dimensional curve

3 – dimensional curve

)('

)('

)('

1)(,

)('

)('

)('

1)(

1

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2

11 tc

tc

tct

tc

tc

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)(')('')(')('')()(

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…tangent, normal

binormal)( normal,)( tangent,)( 321 ttt eee

31)('

)('')(')()(

tc

tctctt

22)('')('

)('''),('')(')()(

tctc

tctctctt

…curvature

…torsion


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

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

Differential geometry & physics

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,...


Pulse wave propagationPulse wave propagation


Ejected blood propagets in the form of the pressure wave

Pulse wave scatteringPulse wave scattering


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 Aorta and major branchingsbranchings


Aortic arch

Diaphragm

Coeliac artery

Mesentric artery

Renalarteries

Abdominalbifurcation

Iliac arteries

Assignment of curvature peaks to Assignment of curvature peaks to mechanical events:mechanical events: cardiac catheterization cardiac catheterization


For comparison we measured three volunteers on the force plate in the same day as they were catheterized.

ResultsResults


InterpretationInterpretation


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. 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


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



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)


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)


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


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


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.


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

detecting targets in human body: what is the common property of radar systems and medical devices?...

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