Transcript What is the common property of radar systems and

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
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
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
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
Summary
What is the output?
What is the common property of radar systems
and medical devices?
Output: multivariate time series
• spatial–temporal character
• data of the form X = S + W
• low signal to noise ratio (SNR)
Signal processing: time series analysis
Targets in human body: processes in the brain,
haemodynamical events,
…
MOTIVATION
Is this a suitable topic for a physicist?
YES !!!
Multivariate time series themselves are analyzed
in physics: geophysics, climatology, meteorology,
astrophysics,…
We exploit mathematical methods commonly used in
quantum mechanics for data processing, namely:
• Differential geometry: quantum waveguides theory
general theory of relativity
• Maximum likelihood estimation: quantum state
reconstruction
• Random matrix theory: quantum billiards
MOTIVATION
Example:
JMA seismic intensity network
Different types of
rock layers filter the
seismic waves.
Aim of data analysis:
• source localization
• earthquake prediction
MOTIVATION
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?
MOTIVATION
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
Force plate data analysis
Postural requirements during quiet standing
- support head and body against gravity
- maintain COM within the base of support
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
Force plate data analysis
Motor strategies (to correct the sway)
Ankle strategy (body = inverted pendulum, vertical forces)
Hip strategy (larger and more rapid, shear forces)
Stepping strategy
Force plate data analysis
Postural control: Central nervous system (CNS)
Spinal cord (reflex, 50 ms)
Brainstem/subcortical (automatic response, 100 ms)
Cortical (voluntary movements, 150 ms)
Cerebellum
-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
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.)
The idea is not new. Ballistocardiography (=usage of
mikromovements for extracting information
on the cardiac activity) is known for more than 70 years.
Cardiovascular dynamics measured by force plate
Cardiac cycle
Total blood circulation:
Veins  right atrium  right ventricle  pulmonary artery  lungs
 pulmonary vein  left atrium  left ventricle  aorta 
branching to capillares  veins
Cardiovascular dynamics measured by force plate
Mechanical activity is triggered by electric one.
Starting point of cycle: ventricle sys. ~ QRS 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 average over cardiac cycles is taken.
Cardiovascular dynamics measured by force plate
Data
Filtering
Averaging
Black box
(Curvatures)
Cardiovascular dynamics measured by force plate
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
Cardiovascular dynamics measured by force plate
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.
Conclusions
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
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.“
Human multiepoch EEG
Experiment:
Human multiepoch EEG
Common property of evoked potentials
and cardiovascular dynamics
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
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
Data
Filtering
Averaging
Black box 1
(MLE)
Black box
(Curvatures)
Black box 2
(Curvatures,
RMT)
Human multiepoch EEG – nonperiodic reversal
Results: channels 57-60
Human multiepoch EEG – nonperiodic reversal
Results: channels 25-28
Human multiepoch EEG – nonperiodic reversal
Results
Human multiepoch EEG – needle sticking
Results
Conclusions
BETTER RESULTS THAN FILTERING/AVERAGING:
• low number of epochs
• low SNR
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
Force plate
  
M  r F


FM
only five independent channels
Usual choice: force components + COP
Mx
My
y
.
x
,
Fz
Fz
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.
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.
Curve: c : a, b  n C n a,b mapping, such that
c' (t )  0, t  [a, b].
Differential geometry & human cardiovascular
dynamics measured by force plate
Frenet 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” animation
click here
Differential geometry & human cardiovascular
dynamics measured by force plate
Assume that c' (t ), c' ' (t ), , c ( n1) (t ) are lin. independent.
The Frenet Frame is the family of orthonormal
vectors {e1 (t ), e 2 (t ), e n (t ) | t [a, b]} called
Frenet
vectors. They are constructed from the derivates of c(t)
using Gram-Schmidt orthogonalization, i. e.
c' (t )
e1 (t ) 
,
c' (t )
e k (t ) 
e k (t )
e k (t )
k 1
, e k (t )  c ( k ) (t )   c ( k ) (t ), ei (t ) ei (t ), k  2, n  1,
e n (t )  e1 (t )  e 2 (t )   e n 1 (t ).
i 1
Differential geometry & human cardiovascular
dynamics measured by force plate
The real valued functions  j (t ), j  1,, n  1 are
called generalized curvatures and are defined as
 j (t ) 
e' j (t ), e j 1 (t )
c' (t )
.
Main theorem of curve theory
Given functions 1 ,  2 ,  ,  n 1 defined on some (a, b) with  j
C n  j 1 - continuous for j  1,  , n  1 and with  j (t )  0
for j  1,  , n  2 and t  (a, b). Then there is unique
(up to Eucleidian transforma tions) n - dimensiona l curve c,
so that c' (t )  1 and c has curvatures
1 ,  2 ,  ,  n 1.
Differential geometry & human cardiovascular
dynamics measured by force plate
2 – dimensional curve
1
e1 (t ) 
c' (t )
c'1 (t ) 
c' (t ),
 2 
 (t )  1 (t ) 
1
e 2 (t ) 
c' (t )
 c'2 (t )
c' (t ) 
 1

 c' '1 (t )c'2 (t )  c' '2 (t )c'1 (t )
c' (t )
3
…tangent,
normal
…curvature
3 – dimensional curve
e1 (t ) tangent, e 2 (t )normal, e3 (t )binormal
 (t )  1 (t ) 
 (t )   2 (t ) 
c' (t )  c' ' (t )
c' (t )
3
…curvature
c' (t )  c' ' (t ), c' ' ' (t )
c' (t )  c' ' (t )
2
…torsion
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 & 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.
Differential geometry & physics
Cardiovascular dynamics measured by force plate
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
Ejected blood propagets in the form of the pressure wave
Cardiovascular dynamics measured by force plate
Pulse 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).
Cardiovascular dynamics measured by force plate
Aorta and major
branchings
Aortic arch
Mesentric artery
Diaphragm
Coeliac artery
Abdominal
bifurcation
Iliac arteries
Renal
arteries
Cardiovascular dynamics measured by force plate
Assignment of curvature peaks to
mechanical events: cardiac catheterization
For comparison we measured three volunteers on the force plate
in the same day as they were catheterized.
Cardiovascular dynamics measured by force plate
Results
Cardiovascular dynamics measured by force plate
Interpretation
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
MLE & human multiepoch EEG
N … spatial channels ,
J … number of epochs
data for j-th epoch:
T … time samples per epoch
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
Full dataset of J epochs: X=[ X1 X2 ... XJ ] ... N x JT matrix
Noise over J epochs: W=[ W1 W2 ... WJ ] ...N x JT matrix
X = [ S S ... S ] + W ,
[ S S ... S ] = HqDT, 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=1)
R = WWT … 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(2pft), cos(2pft) 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=1 (we take all
frequencies)
MLE & human multiepoch EEG
1. Univariate normal distribution
normally distributed random quantity x has pdf:
 ( x   )2 
1

f ( x | , ) 
exp  
2
2
 2


where m is the mean and s2 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 aTX is univariate normal.
m=[
E(X1) ... E(Xm) ]T
Covariance matrix: S=E
m)T]
[( X – m)(X-
Mean:
Theorem: If X is normally distributed then the pdf
function is
1
 1
T 1

f ( X |  , ) 
exp   ( X   )  ( X   ) 
m/ 2
det  2 
 2

MLE & human multiepoch EEG
3. Normal distribution for multivariate time series
Under all above assumptions, the pdf can be written as
f ( X | R, , H ) 


 1
1
T
T T 
exp

Tr
R
(
X

H

D
)(
X

H

D
) 

NTJ / 2
TJ / 2
(det R) 2 
 2

1
Thus, we are looking for unknown matrices R, q
and H to maximize the likelihood function for our data
X.
1
 1

l ( R, , H | X ) 


1
T
T T
exp

Tr
R
(
X

H

D
)(
X

H

D
) 

NTJ / 2
TJ / 2
(det R) 2 
 2

It was done by Baryshnikov et al.
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.