Electrical Impedance Tomography: Image Algorithms …€¦ · A.Adler, Mar 2008 Electrical...
Transcript of Electrical Impedance Tomography: Image Algorithms …€¦ · A.Adler, Mar 2008 Electrical...
A.Adler, Mar 2008 Electrical Impedance Tomography 1
BIOM5200 – Medical Imaging
Electrical Impedance
Tomography:
Image Algorithms and
Applications
Andy AdlerAssistant Professor, SCE, Carleton U
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Outline
• Electrical Impedance Tomography
• Applications
• Physics
• Image Reconstruction
• Future Work
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Electrical Impedance Tomography
• Relatively new medical imaging
technique (early 1990’s)
• Body Surface Electrodes apply current
patterns and measure the resulting
voltages
• Distribution of conductivity is calculated
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EIT: Block Diagram
+
+
CurrentSource
Amplifiers
Controler
Dataacquisition
Medium
+
Imaging SystemΩ
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Electrode placement to monitor
the lungs and heart
Adult Preterm infant
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EIT: Applications
EIT can image/monitor processes involving movement of conductive fluids and gasses
• Lungs
• Heart / perfusion (blood flow)
• GI tract
• Brain
• Breast
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Application: Breathing
Chest images of tidal breathing in normal
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Application: Heart Beat
EIT signal in ROI around heart and ECG
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Why image lungs?
Respiratory FailureInadequate gas exchange by the respiratory system.
Hypoxemia PaO2 < 60 mmHg or Hyercapnia PaCO2 > 45 mmHg
Causes
• Pulmonary dysfunction – Asthma ,Emphysema , Chronic obstructive airway disease,
Pneumonia , Pneumothorax, Hemothorax, Acute Respiratory Distress Syndrome (ARDS), Cystic Fibrosis
• Cardiac dysfunction – Pulmonary edema, Arrhythmia, Congestive heart failure, Valve
pathology
Treatment– Emergency treatment: cardiopulmonary resuscitation.
– Treatment of the underlying cause is required.
– Mechanical ventilation may be required.
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Mechanical Ventilationused in acute settings (ICU). Often a life-saving technique, but has
many complications – pneumothorax,
– airway injury,
– alveolar damage,
Accordingly it is generally weaned off or to minimal settings assoon as possible.
Positive pressure in contrast to the more historically common negative pressure ventilators sucking air into the lungs.
Iron Lung
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Modes of Ventilation
classifications based on how to control the ventilator breath.
• Breath termination
• Breath initiation
• High Frequency Ventilation (HFV)
As microprocessors are incorporated into ventilator design, ventilators use combinations of all modes and flow-sensing
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Why image lungs?PneumoniaA: Normal chest x-ray.
B: Abnormal chest x-ray
shadowing from
pneumonia in the right lung
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FRC
Static Mechanics: ventilation
Stiffer lung
Stiffer lungs have decreased
resting volume (FRC)
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Regional ventilation
Data from
Frerichs et al (2003)
Intensive Care Med..
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No change in
lower lungs
until pressures
get really big
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Applications: Brain
Applications
• Hemorrhage
• Localization of epileptic foci
Newborn with EIT electrode cap on head
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Industrial Applications
Process Tomography
• Fluid/gas flow in pipes
• Metal Castings
Geophysics
• Undersurface geology
• Mine detection
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EIT: Advantages
EIT is a relatively low resolution imaging
modality, with several advantages
• Non-invasive
• Non-cumbersome
• Suitable for monitoring
• Underlying technology is low cost
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Non-invasive
0.1
1
10
100
30 100 300 1K 3K 10K 30K
Frequency (Hz)
Current
(mA)
Percentile 99
Percentile 50
Percentile 1
EIT system
Thresholds for cutaneous perception of electric current vs. frequency and EIT system
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Hardware: Electrodes
• Current stimulation is better than voltage, because it accounts for electrode contact impedance
• Traditionally EIT uses adjacent current drive.
• Some systems separate drive and measurement electrodes, using adaptive current patterns
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EIT: Physics
• Within medium Ω there is E and J.
•
dt
dod
c
EJ
EJ
εε
σ
=
=
( )EJ ojωεεσ −=
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EIT: Physics
In the absence of magnetic fields
No charge build up in conductive medium
We have
V−∇=E
0J =−=•∇t∂
∂ρ
( ) Ω=∇−•∇ in 0Vj oωεεσ
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EIT: Physics
Current is applied at electrodes
Body need to be grounded, somewhere
eIt
=−=•∇∂
∂ρJ
point someat 0=V
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EIT: Numerical Models
In order to calculate measurements from
conductivities, we can use:
• Analytic Techniques
– Analytic models exist for elliptic 3D media; however, numerical approximations of sums required
• Numerical Models
– Finite Element Techniques, main method
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Finite Element Models
Simple Model
with 64 elements
Used for inverse solution
-1 -0.5 0 0.5 1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
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Model of Borsic, Physiol Meas, 22:77-83, 2002
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-1-0.5
00.5
1
-1-0.5
00.5
1
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
Finite Element Models
“Simple” 3D Model with
768 elements
Used for inverse solution
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Image Reconstruction:Static Imaging
Static imaging reconstructs the absolute
conductivity from measurements.
Algorithms:
• Iterative (Newton-Raphson)
• Layer Stripping
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CurrentInjection
Patient
and
EIT datameasurement
Real Data
Finite Element Model
EIT data
simulation Simulation Data
Does simulationapproximate
real data
Finishedyes
Update
conductivity
distribution
no
Iterative (Absolute) Image
Reconstruction
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Absolute Imaging Difficulties
• Extremely sensitive to uncertainties in
electrode position
– Need to know where electrodes are to and electrode shape to 1mm
– “Absolutely” must do 3D
• Numerical instability
• Slow reconstructions
• Is muscle in chest isotropic?
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Difference
Imaging: Example
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Difference Imaging
• Calculate ∆ conductivity
from ∆ measurements
• Inverse problem linearized
• reduced sensitivity to electrode and
hardware errors.
• Suitable for physiological imaging: lung,
heart, GI
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Inverse Techniques
• We can pose dynamic imaging as linear
inverse, using a sensitivity matrix
j
jhhj
δ
δσσ )()( +−=
zzz
σ∆= Hz
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Inverse Techniques
• Classic least-squares inverse
( ) zHHHx
Hxz
tt 1ˆ
−=
=
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Model based matrix inverses
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Matrix Techniques
However, problem is:
• ill-conditioned: measurements depend much more on data near electrodes
than in centre
• ill-formed: more unknowns than
measurements
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Regularized Imaging
Handwaving argument for regularization:
used for ill-posed and ill-formed problems to find a solution with:
• Low error: small ( z – Hx )
• Stable: small change in x for small ∆z
• Good looking:
– Somewhat hard to define, but includes
smoothness, clean edges, etc.
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MAP estimates
• MAP approach says choose x such that
f(x|z) is maximized
– In other words, choose the image that is most likely, considering the measured data
• Bayes Rule
( ) ( ) ( )( )z
xxzzx
f
fff =
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MAP estimates
f(z|x) the distribution of measurements given an image– Based on forward model and noise
properties
f(z) distribution of measurements– Not a parameter of MAP estimate
f(x) distribution of image– Based on a priori knowledge of physically
possible and likely images distributions
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Regularized Imaging
Given Linear Model:
Maximum A Posteriori (MAP) estimate is:
( ) ( )∞−−−−− ++=
+=
xRRHRHRHx
nHxz
11111ˆ xnt
xnt
z
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Image Reconstruction
• Forward Model (linearized)
= ×
Measurements(difference)
Image(difference)
Jacobian
System is underdetermined
+ noise
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Image Reconstruction
Regularized linear Inverse Model
– ×
Norm weighted by
measurementaccuracy
+ PenaltyFunction
2
Way to introduce
“prior knowledge”into solution
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Regularized Imaging
• Parameters Rx, Rn, x∞ , represent a priori
statistical knowledge of problem
[ ]
( ) ( )[ ] [ ]
[ ]
==
−==
=
∞∞∞∞
∞
OM
L
22
21
0
0
σ
σ
nnR
xxxxx-xx-xR
x x
n
x
t
ttt
E
EE
E
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Choice of parameter Rx
• Parameter is a “penalty function”
• Many regularization approaches use a diagonal matrix– Tikhonov regularization uses the scaled
identity matrix
– This will penalize large amplitude pixels in image
• We choose a dense matrix– Penalize image frequency content above
maximum possible with measurements
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Regularization: Hyperparameters
Regularizations techniques must finally
introduce a “hyperparameter” (µ)
( ) ( )
nscorrelatio image relative the ie. ,
amplitudes noise relative the ie. ,
1
1
−
−
∞
−
=
=
++=
x
n
RQ
RW
QxWzHQWHHx
2
2
1
1
1
ˆ
x
n
tt
where
σ
σ
µµ
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Regularization: Hyperparameters
µ is thus the ratio of image and noise
amplitudes,
it can be interpreted as a filter noise figure
2
2
y
x
σ
σµ =
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Regularized Inverse
Parameters:
• W: models measurement noise
• Q: penalizes image features which are greater than data supports
• : represents the background conductivity distribution (heart,lungs,etc)
• µ: “hyper-parameter” amount of regularization
∞x
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Advantages of Regularization
• Stabilizes ill-conditioned inverse
• Introduction of a priori information
• Control of resolution-noise performance
trade-off
• MAP inverse justifies the formulation in
terms of Bayesian statistics
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Noise – Resolution Tradeoff
LittleRegularization
(small penalty)
No Noise
-3dB
SNR
Lots of
Regularization(large penalty)
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Applications …
• Electrode Errors
• Electrode Movement
• 3D Imaging / Electrode Placement
• Temporal Filtering
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Electrode Measurement Errors
Experimental measurements with EIT quite often show large errors from
one electrode
Causes aren’t always clear
– Electrode Detaching
– Skin movement
– Sweat changes contact impedance
– Electronics Drift?
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Example of electrode errors
Images measured in anaesthetised, ventilated dog
A. Image of 700 ml ventilation
B. Image of 100 ml saline instillation in right lung
C. Image of 700 ml ventilation and 100 ml saline
A B C
“Bad”
Electrode
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Measurements with “bad” electrode
“Bad”Electrode
2
3
4
5
6
7
0
1
7067564534231201
XX∗∗∗∗∗∗∗∗X70
XXX∗∗∗∗∗∗∗∗67
XXX∗∗∗∗56
∗∗∗∗∗∗∗∗XXX∗∗∗∗∗∗∗∗∗∗∗∗45
∗∗∗∗∗∗∗∗∗∗∗∗XXX∗∗∗∗∗∗∗∗34
∗∗∗∗XXX23
∗∗∗∗∗∗∗∗XXX12
X∗∗∗∗∗∗∗∗XX01
∗∗∗∗ “bad” measurement
X measurement at current injection
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“Zero bad data” solution
“Traditional solution” (in the sense that
I’ve done this)
– ×
2
σ1
σ2
σ3
σ4
-2
-2
-2
-2
Error Here
Replace
With zero
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Regularized imaging solution
Electrode errors are large measurement
noise on affected electrode
– ×
2
σ1
σ2
σ3
σ4
-2
-2
-2
-2
Error Here
Low
SNR
here
Replace
With zero
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Simulation
Data simulated with 2D FEM with 1024 elements
– not same as inverse model
“Bad”Electrode
Position in % of medium diameter
Small
targets simulated at
different
radial positions
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Simulation results for opposite driveNo Electrode Errors
Zero Affected Measurements
Regularized Image
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How does this work with real data?
A. Image of 700 ml ventilation
B. Image of 100 ml saline instillation in right lung
C. Image of 700 ml ventilation and 100 ml saline
A B C
“Bad”
Electrode
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Electrode Movement
Electrodes move
• with breathing
• with posture change
Simulations show broad central artefact in images
A B
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Imaging Electrode Movement
• Forward model image includes
movement
= ×
Jacobiannow includes
measurement changedue to movement
+ noise
“image” nowincludes
x,y sensormovement
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Images of electrode movement
Simulation: tank twisted in 3D
y x
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Top
slice
Middle
slice
Bottom
slice
Simulation Standard
Algorithm
with electrode
movement
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EIT makes fast measurements.
Can we use this fact?
……
0-1 +2 +n+1-2-n
past now future
=
Jacobian
…
Image sequenceMeasurement
sequence
0-1 +2 +n+1-2
past now future
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Temporal Reconstruction
Temporal Penalty Functions
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
likely quite likely unlikely
Standard EIT approaches to not take this into account
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GN vs. Temporal Inverse
Gauss-Newton solver
Solve time = 5.33 s
(with caching) = 0.22 s
Temporal solver
(4 time steps)Solve time = 34.81 s
(with caching) = 0.60 s
1. Noise free data (IIRC tank)
2. Data with added 6dB SNR noise
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Gauss Newton vs. Temporal
Inverse (6db SNR)
Gauss-Newton solver
Solve time = 5.33 s(with caching) = 0.22 s
Temporal solver(4 time steps)
Solve time = 34.81 s(with caching) = 0.60 s