Transcript Biomedical Imaging I

Biomedical Imaging 2
Class 2 – Diffuse Optical Tomography (DOT)
01/22/08
BMI2 SS08 – Class 2 “DOT Theory” Slide 1
Acknowledgment
Dr. Ronald Xu
Assistant Professor
Biomedical Engineering Center
Ohio State University
Columbus, Ohio
Slides 11, 14-18, 21 and 22 in this presentation were created by Prof. Xu, and can be found in their
original context at the following URL:
http://medimage.bmi.ohio-state.edu/resources/medimage_ws2005_Xu-image_workshop_2.16.05.ppt
BMI2 SS08 – Class 2 “DOT Theory” Slide 2
What Are We Measuring?
Input (source): s(rs,Ωs)
Output (measurement):
d(rs,Ωs;rd,Ωd)
Constitutive property/ies
(contrast): x(ri[,Ωi])
Transfer function: T(ri,Ωi)
= T(x(ri[,Ωi]))
BMI2 SS08 – Class 2 “DOT Theory” Slide 3
What Are We Measuring?
Input (source): s(rs,Ωs)
Output (measurement):
d(rs,Ωs;rd,Ωd)
Constitutive property/ies
(contrast): x(ri[,Ωi])
Transfer function: T(ri,Ωi)
= T[x(ri[,Ωi])]
r
dV = (dr)3
0
BMI2 SS08 – Class 2 “DOT Theory” Slide 4
More on Transfer Function
•
Strictly speaking, is a mathematical operator, not a function
– Maps one function into another function
• Familiar examples: d/dx; multiply by x and add 2; ∫dx (i.e., indefinite integral)
– Different from a function (maps a number into another number) or a
functional (maps a function into a number)
• Strictly speaking, a –function is actually a functional.
•
T{s}  d
– If medium is linear, then:
d  rs , Ωs  ; rd , Ωd    
ri


3
T
r
,

s
r
,
Ω
d

d
r






s 
s 
Ω i i
i
• i.e., overall effect of entire volume of material on the input is the summation
of each volume element’s individual effects
– Nonlinearity makes problem of determining x(r) far more difficult
– We’re not home free even if medium is linear, given the dependence of
T on x.
BMI2 SS08 – Class 2 “DOT Theory” Slide 5
When Can We Solve for x(r)?
• Most generally, T is influenced by x
• Most tractable case: T is medium-independent
– i.e., T(x) = T0·x, or T(x) = T0·f(x).
• Also sometimes doable: T is not medium-independent,
but can be treated as if it were, for the purpose of
computing a successive approximation sequence:
– T0  x1  T1  x2  T2  x3  ...
• In retrospect, it is easy to see why some types of
medical imaging were successfully developed long
before others, and why some produce higher–resolution
images than others.
BMI2 SS08 – Class 2 “DOT Theory” Slide 6
x–ray CT — Tractable or Not?
Because we exclude the scattered photon component
from the detectors, we have T0 = –functions, and
f(x) = f(μ) = e -μ
BMI2 SS08 – Class 2 “DOT Theory” Slide 7
Nuclear Imaging — Tractable or Not?
Besides collimation, we also have to deal with the
attenuation phenomenon, which makes the problem
non–separable
Successive approximation strategies have been
employed with some success.
BMI2 SS08 – Class 2 “DOT Theory” Slide 8
Ultrasound CT — Tractable or Not?
Successive approximation strategy can be successfully employed when
spatial variation of the acoustic impedance is weak.
For highly heterogeneous (scattering) media, ultrasound CT may be
possible if we can apply either the Born (i.e., negligible variation in
ultrasound wave amplitude within scattering objects) or Rytov (i.e.,
negligible variation in ultrasound wave phase within scattering objects)
approximation.
BMI2 SS08 – Class 2 “DOT Theory” Slide 9
An Intractable Case
Object (tissue) is
illuminated with near
infrared (NIR) light (i.e.,
wavelengths between 750
nm and 1.2 μm). (What
is photon energy?)
The light spreads out in all
directions from the point
of illumination, similar to a
droplet of ink in water
diffusing away from its
initial location.
1) Is T strongly (and nonlinearly) dependent on x in
this case? 2) What constitutes x?
BMI2 SS08 – Class 2 “DOT Theory” Slide 10
How Photons Interact with Biological Tissue
Scattered and reflected
s

s’
Scattered and
absorbed
mal, msl, g
Scattered and transmitted
BMI2 SS08 – Class 2 “DOT Theory” Slide 11
Quantitative Assessment of Absorption and Scattering
1. Inner surfaces are
coated with a bright,
white, highly reflective
material (very high µs,
very low µa)
Detector
2. Eventually, all nonabsorbed photons are
captured by one or
another of the detectors
[From: J. W. Pickering, S. A. Prahl, et al., “Double-integrating-sphere
system for measuring the optical properties of tissue,” Applied Optics
32(4), 399-410 (1993).]
3. An upper limit on the
sample material’s µa can
be computed from the
difference
between
incident and detected
light levels
BMI2 SS08 – Class 2 “DOT Theory” Slide 12
Quantitative Assessment of Absorption and Scattering
1. Inner surfaces are
coated with a dark,
matte, highly absorptive
material (very high µa,
very low µs)
Detector
2. Detector receives
photons that are not
removed
from
the
incident beam, by either
absorption or scattering
3. So, measuring the
decrease of detected
light
as
the
slice
thickness
increases
gives an estimate of the
sum µa + µs
BMI2 SS08 – Class 2 “DOT Theory” Slide 13
Scattering is Caused by Tissue Ultrastructure
(http://omlc.ogi.edu)
BMI2 SS08 – Class 2 “DOT Theory” Slide 14
Absorption is Caused by Multiple Chromophores
BMI2 SS08 – Class 2 “DOT Theory” Slide 15
In NIR Region, Hb and HbO are Major Sensitive Absorber
4000
- Deoxy-hemoglobin
l2
l1
 HbO
m al1   HbO
m al 2
 Hb  l1 l 2 l 2 l1
 Hb HbO   Hb  HbO
extinct coeff (cm-1/mol/liter)
3500
- Oxy-hemoglobin
3000
2500
l1 l 2
l 2 l1
 Hb
m a   Hb
m
 HbO  l1 l 2 l 2 la1
 Hb HbO   Hb  HbO
l1 = 690nm
2000
l2 = 830nm
1500
[ HbT ]  [ Hb]  [ HbO ]
1000
SO2 
500
650
700
750
800
850
[ HbO ]
[ HbT ]
900
wavelength (nm)
BMI2 SS08 – Class 2 “DOT Theory” Slide 16
What Near Infrared Light Can Measure?
• Absorption measurement
–
–
–
–
–
Tissue hemoglobin concentration
Tissue oxygen saturation
Cytochrome-c-oxidase concentration
Melanin concentration
Bilirubin, water, glucose, …
• Scattering measurement
–
–
–
–
Lipid concentration
Cell nucleus size
Cell membrane refractive index change
…
BMI2 SS08 – Class 2 “DOT Theory” Slide 17
Why Tissue Oximetry?
• Tissue oxygenation and hemoglobin concentration
are sensitive indicators of viability and tissue
health.
• Many diseases have specific effects on tissue
oxygen and blood supply: stroke, vascular
diseases, cancers, …
• Non-invasive, real time, local measurement of
tissue O2 and HbT is not commercially available
BMI2 SS08 – Class 2 “DOT Theory” Slide 18
Why do we want to know μa?
μa = μa(Hb-oxy) + μa(Hb-deoxy) + μa(H2O) + μa(lipid)
+ μa(cyt-oxidase) + μa(myoglobin) + …
μa(X, λ) = ε(X,λ)∙[X]
Concentration of
X (M, mol-L-1)
Absorption
coefficient of X
(cm-1)
Molar extinction
coefficient
(cm-1M-1)
BMI2 SS08 – Class 2 “DOT Theory” Slide 19
Why do we want to know μa?
μa = μa(Hb-oxy) + μa(Hb-deoxy) + μa(H2O) + μa(lipid)
+ μa(cyt-oxidase) + μa(myoglobin) + …
Rule: To get quantitatively accurate chromophore
concentrations, the number of distinct wavelengths
used for optical imaging must be at least as large
as the number of compounds that contribute to the
overall μa
BMI2 SS08 – Class 2 “DOT Theory” Slide 20
Why Near Infrared? Pros and Cons Compared with Other Imaging Modalities
• Advantages:
–
–
–
–
–
–
–
–
Deep penetration into biological tissue
Non-invasive
Non-radioactive
Real time functional imaging
Portable
Low cost
Tissue physiological parameters
Potential of molecular sensitivity
• Disadvantages:
– Low spatial and depth resolution
– Hard to quantify
BMI2 SS08 – Class 2 “DOT Theory” Slide 21
Near Infrared Diffuse Optical Imaging: Problem Definition
• Find embedded tissue heterogeneity
• By solving: OD  ln fo  f (St O2B , H bt B , StO2T , H bt T )
fi
source
fi
fo
detector
StO2B, HbtB
StO2T
HbtT
BMI2 SS08 – Class 2 “DOT Theory” Slide 22
Continuous Wave (C.W.) Measurements
• Simplest form of OT: lowest spatial resolution, “easy”
implementation, greatest penetration
– Measuring transmission of constant light intensity (DC)
– Simple, least expensive technology  most S-D pairs
– High “frame rates” possible
BMI2 SS08 – Class 2 “DOT Theory” Slide 23
Example: Optical brain imaging
• “Partial view” or back reflection geometry
Source / Detector 1
Detector 2
2-3 cm
Detector 3
Scalp
Bone
CSF
Cortex
BMI2 SS08 – Class 2 “DOT Theory” Slide 24
Time-Resolved Measurements
Prompt or ballistic Photons (t = d/c)
I
“Snake” Photons
I
Diffuse Photons
t
t0
t
d
t0
• Measuring the arrival time/temporal spread of short pulses (<ns) due
to scattering & absorption (narrowing the “banana”)
• Expensive, delicate hardware (single-photon counters, fast lasers,
optical reflections, delays…)
• Long acquisition times (low frame rates)
• Potentially better spatial resolution than DC measurements
BMI2 SS08 – Class 2 “DOT Theory” Slide 25
Frequency-Domain Measurements
I
I
t
t0
Photon density waves
t
t0
I
I
Modulation
Phase
t
t
t0
t0
•
Propagation of photon density waves (PDW): lPDW = 9 cm, cPDW = 0.06 c (*
•
Measure PDW modulation (or amplitude) and phase delay
•
RF equipment (100MHz-1GHz)
•
Wave strongly damped, challenging measurement
(*
f = 200 MHz, μa = 0.1 cm-1, μs = 10 cm-1 n = 1.37
BMI2 SS08 – Class 2 “DOT Theory” Slide 26
Theoretical Descriptions of NIR Propagation Through Tissue
•
Quantum Electrodynamics
•
Classical Electrodynamics (Maxwell’s equations)
•
Radiation Transport Equation
1 ¶  r , ,t 
 S  r , ,t       r , ,t    ma  r   ms  r    r , , t 
c
¶t
source
streaming
absorption and
scattering out
 ms  r   f       r , ,t  d 
4
scattering in
•
Diffusion Equation
1 ¶f r ,t 
  D  r  f r ,t    ma  r  f r ,t   S  r ,t 
c ¶t
– Assumes (among other things) that μs(r) >> μa(r).
BMI2 SS08 – Class 2 “DOT Theory” Slide 27
Making the Problem Tractable — Perturbation Strategy
•
For a medium of known properties x 0(r) = {μa0(r), D 0(r)}, we can find the
transfer function to any desired degree of accuracy: T(x 0){s} = d 0.
– We will refer to the above as our reference medium.
•
What if an (unknown) target medium is different from the reference medium
by at most a small amount at each spatial location?
– i.e., μa(r) = μa0(r) + Δμa(r), |Δμa(r)| << μa0(r);
D (r) = D 0(r) + ΔD (r), |ΔD(r)| << D 0(r).
– Δμa(r) = absorption coefficient perturbation, ΔD(r) = diffusion coefficient
perturbation
•
Then the resulting change in d is approximately a linear function of the
coefficient perturbations
– i.e.,
d  d 0    ¶d ¶ma  m
V

d  d 0  d 
0
a
V
,D
0
ma   ¶d ¶D  m 0 ,D 0 D  d 3r ,
a

Wa ma  WD D  d 3r
Wa    ¶d ¶ma  m
0
a
,D 0
, WD    ¶d ¶D  m 0 ,D 0
a
BMI2 SS08 – Class 2 “DOT Theory” Slide 28
Making the Problem Tractable — Perturbation Strategy II
• In practice, medium is divided into a finite number N of pixels
(“picture element” – 2D imaging) or voxels (“volume element” –
3D imaging)
• We further assume that each element is sufficiently small that
there is negligible spatial variation of μa or D within it.
• Integral in preceding slide becomes a sum:
N

d   Wan man  WDn D n
n 1
Wan    ¶d ¶ma  m
0
a
,D 0

V n , WD    ¶d ¶D  m 0 ,D 0 V n
a
• Perturbation equations for all source/detector combinations are
combined into a system of linear equations, or matrix
equation.
BMI2 SS08 – Class 2 “DOT Theory” Slide 29
Making the Problem Tractable — Perturbation Strategy III
• Perturbation equations for all source/detector
combinations are combined into a system of linear
equations, or matrix equation:
d   Wa
 d 1  Wa11 Wa12
 d   21
22
W
W
2
a
a



 

  M1

d
WaM 2
 M  Wa
 μa 
WD  


D


Wa1N WD11
Wa2N WD21
WD12
WD22
WaMN WDM 1 WDM 2
 ma1 
 2 
 ma 

WD1N  
N 
2N  
WD   ma 
  D 1 


MN 
2
WD   D 




N
 D 
BMI2 SS08 – Class 2 “DOT Theory” Slide 30
Dilemma:
Many different combinations of μa and μs are
consistent with any given non-invasive light
intensity measurement
log10(Intensity)
0.01
-1
0.02
-2
0.03
0.04
-3
μa 0.05
-4
0.06
0.07
-5
0.08
-6
0.09
0.1
-7
2
4
μs
6
8
10
BMI2 SS08 – Class 2 “DOT Theory” Slide 31
Solution, Part 1:
Few spatial distributions of μa and μs are
consistent with many nearly simultaneous noninvasive light intensity measurement
(Cavernous
hemangioma)
BMI2 SS08 – Class 2 “DOT Theory” Slide 32
Solution, Part 2:
Simplify mathematical problem by introducing an
additional light-scattering medium into the mix
=
-
The problem of deducing
the spatial distributions of μa
and μs in this medium, from
light intensity measurements around its border, is
very difficult
Figuring out the difference
between the spatial distributions of μa and μs in these
two media is much easier
BMI2 SS08 – Class 2 “DOT Theory” Slide 33
Solution, Part 2:
As a practical matter, most useful method is to use
a spatially homogeneous second medium (i.e.,
reference medium)
µs = 9 cm-1
µs = 9 cm-1
µa = 0.05 cm-1 µa = 0.07 cm-1
µs = 10 cm-1
µs = 11 cm-1
µa = 0.06 cm-1
µs = 11 cm-1
µa = 0.05 cm-1 µa = 0.07 cm-1
Δµs = -1 cm-1
Δµs = -1 cm-1
Δµa = -0.01 cm-1 Δµa = 0.01 cm-1
Δµs = 1 cm-1
Δµs = 1 cm-1
Δµa = -0.01 cm-1 Δµa = 0.01 cm-1
BMI2 SS08 – Class 2 “DOT Theory” Slide 34
Solution, Part 3:
Linear perturbation strategy for image reconstruction
µs = 10 cm-1
µa = 0.06 cm-1
Use a computer simulation (or
a homogeneous laboratory
phantom) to derive the pattern
of light intensity measurements
around the reference medium
boundary
Additional computer simulations
determine the amount by which
the detected light intensity will
change, in response to a small
increase (perturbation) in μa or μs
in any volume element (“voxel”)
BMI2 SS08 – Class 2 “DOT Theory” Slide 35
Solution, Part 3:
Linear perturbation strategy for image reconstruction
Each
of
these
shades
of
gray
represents a different
number. Let’s write
them all as a row
vector.
Because increasing
μa decreases the
amount of light that
leaves the medium
One
number
(weight) for each
voxel
BMI2 SS08 – Class 2 “DOT Theory” Slide 36
Solution, Part 3:
Linear perturbation strategy for image reconstruction
Repeat process just described, for all
source-detector combinations.
WEIGHT
matrix
BMI2 SS08 – Class 2 “DOT Theory” Slide 37
Solution, Part 3:
Linear perturbation strategy for image reconstruction
Measurement perturbation (difference) is
directly proportional to interior optical
coefficient perturbation. Weight matrix gives
us the proportionality.
µs = 9 cm-1
µs = 9 cm-1
µa = 0.05 cm-1
µa = 0.07 cm-1
µs = 11 cm-1
µs = 11 cm-1
µs = 10 cm-1
µa = 0.06 cm-1
R  WX
µa = 0.05 cm-1
µa = 0.07 cm-1
Δµs = -1 cm-1
Δµs = -1 cm-1
Δµa = -0.01 cm-1 Δµa = 0.01 cm-1
Δµs = 1 cm-1
Δµs = 1 cm-1
Δµa = -0.01 cm-1 Δµa = 0.01 cm-1
BMI2 SS08 – Class 2 “DOT Theory” Slide 38
Solution, Part 3:
Linear perturbation strategy for image reconstruction
Reconstructing image of μa and μs boils down to
solving a large system of linear equations.
∆R and W are known, and we solve for the
unknown ∆X
Formal mathematical term for this is inverting the
weight matrix W.
1
R  WX  X  W R
BMI2 SS08 – Class 2 “DOT Theory” Slide 39
Real-world Issue 1:
Coping with noise (random error) in clinical
measurement data
Linear system solutions are additive:
W
 R1  R 2   W
1
1
1
R1  W R 2
 X1  X 2
W
 R     W
1
Noise in data
1
R  W

1
 X  E
Noise image
BMI2 SS08 – Class 2 “DOT Theory” Slide 40
Real-world Issue 1:
Coping with noise (random error) in clinical
measurement data
In practice it can easily happen that E is
larger than ∆X.
To suppress the impact of noise,
mathematical
techniques
known
as
regularization are employed.
BMI2 SS08 – Class 2 “DOT Theory” Slide 41
w a  s m , d n ; x, y  
¶ Rsm ,dn
¶ ma x ,y
ms  10 cm-1, ma  0.06 cm-1
 D  1/ 3  ms  ma   0.0331 cm
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w D  s m , d n ; x, y  
¶ Rsm ,dn
¶ Dx ,y
w s  s m , d n ; x, y  
¶ Rsm ,dn
¶ ms x ,y

¶ Rsm ,dn ¶ Dx ,y
¶ Dx ,y ¶ ms x ,y
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Physical diameter /
thickness
(8 cm)(10.06 cm-1) = 80.48
Optical diameter /
thickness
Total attenuation
coefficient
BMI2 SS08 – Class 2 “DOT Theory” Slide 62