Date post: | 02-Jan-2016 |
Category: |
Documents |
Upload: | david-carson |
View: | 213 times |
Download: | 0 times |
BMI2 SS07 – Class 2 “DOT Theory” Slide 1
Biomedical Imaging 2Biomedical Imaging 2
Class 2 – Diffuse Optical Tomography (DOT)
01/23/07
BMI2 SS07 – Class 2 “DOT Theory” Slide 2
AcknowledgmentAcknowledgment
Dr. Ronald XuDr. Ronald XuAssistant ProfessorAssistant Professor
Biomedical Engineering CenterBiomedical Engineering CenterOhio State UniversityOhio State University
Columbus, OhioColumbus, 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 SS07 – Class 2 “DOT Theory” Slide 3
What Are We Measuring?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 SS07 – Class 2 “DOT Theory” Slide 4
What Are We Measuring?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])] 0
rdV = (dr)3
BMI2 SS07 – Class 2 “DOT Theory” Slide 5
More on Transfer FunctionMore 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:
• 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.
3, ; , , ,i i
s s d d i i s sd T r s d d rr Ω
r Ω r Ω r Ω
BMI2 SS07 – Class 2 “DOT Theory” Slide 6
When Can We Solve for x(r)?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 SS07 – Class 2 “DOT Theory” Slide 7
x–ray CT — Tractable or Not?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 SS07 – Class 2 “DOT Theory” Slide 8
Nuclear Imaging — Tractable or Not?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 SS07 – Class 2 “DOT Theory” Slide 9
Ultrasound CT — Tractable or Not?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 SS07 – Class 2 “DOT Theory” Slide 10
An Intractable Case An Intractable Case
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?
Object (tissue) is illuminated with near infrared (NIR) light (i.e., wavelengths between 750 nm and 1.2 μm). (What is photon energy?)
BMI2 SS07 – Class 2 “DOT Theory” Slide 11
How Photons Interact with Biological Tissue
s s ’
Scattered and reflected
Scattered and absorbed
Scattered and transmitted
a, s
, g
BMI2 SS07 – Class 2 “DOT Theory” Slide 12
[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).]
Detector
1. Inner surfaces are coated with a bright, white, highly reflective material (very high µs, very low µa)
2. Eventually, all non-absorbed photons are captured by one or another of the detectors
3. An upper limit on the sample material’s µa can
be computed from the difference between incident and detected light levels
Quantitative Assessment of Absorption and Scattering
BMI2 SS07 – Class 2 “DOT Theory” Slide 13
Detector
1. Inner surfaces are coated with a dark, matte, highly absorptive material (very high µa, very low µs)
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
Quantitative Assessment of Absorption and Scattering
BMI2 SS07 – Class 2 “DOT Theory” Slide 14
Scattering is Caused by Tissue Ultrastructure
(http://omlc.ogi.edu)
BMI2 SS07 – Class 2 “DOT Theory” Slide 16
In NIR Region, Hb and HbO are Major Sensitive Absorber
extin
ct c
oeff
(cm
-1/m
ol/l
iter)
wavelength (nm)
650 700 750 800 850 900
500
1000
1500
2000
2500
3000
3500
4000- Deoxy-hemoglobin
- Oxy-hemoglobin
1 = 690nm
2 = 830nm
2 1 1 2
1 2 2 1
1 2 2 1
1 2 2 1
Hb
HbO
HbO a HbO a
Hb HbO Hb HbO
Hb a Hb a
Hb HbO Hb HbO
2
[ ] [ ] [ ]
[ ]
[ ]
HbT Hb HbO
HbOSO
HbT
BMI2 SS07 – Class 2 “DOT Theory” Slide 17
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 SS07 – Class 2 “DOT Theory” Slide 18
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 SS07 – 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) + …
μa(X, λ) = ε(X,λ)∙[X]
Concentration of X (M, mol-L-1)
Molar extinction coefficient(cm-1M-1)
Absorption coefficient of X(cm-1)
BMI2 SS07 – Class 2 “DOT Theory” Slide 20
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
μa = μa(Hb-oxy) + μa(Hb-deoxy) + μa(H2O) + μa(lipid)
+ μa(cyt-oxidase) + μa(myoglobin) + …
Why do we want to know μa?
BMI2 SS07 – Class 2 “DOT Theory” Slide 21
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 SS07 – Class 2 “DOT Theory” Slide 22
StO2B, HbtB
StO2T
HbtT
StO2B, HbtB
StO2T
HbtT
Near Infrared Diffuse Optical Imaging: Problem Definition
• Find embedded tissue heterogeneity• By solving:
i
source
o
detector
2 2ln ( , , , )B B T Tot b t b
i
OD f S O H t S O H t
BMI2 SS07 – Class 2 “DOT Theory” Slide 23
Continuous Wave (C.W.) MeasurementsContinuous 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 SS07 – Class 2 “DOT Theory” Slide 24
Example: Optical brain imagingExample: Optical brain imaging
• “Partial view” or back reflection geometry
Scalp
Bone
Cortex
CSF
2-3 cm
Source / Detector 1
Detector 2
Detector 3
BMI2 SS07 – Class 2 “DOT Theory” Slide 25
Time-Resolved MeasurementsTime-Resolved Measurements
• 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
t
I
t0 t
I
t0
Prompt or ballistic Photons (t = d/c)
d
“Snake” Photons
Diffuse Photons
BMI2 SS07 – Class 2 “DOT Theory” Slide 26
Frequency-Domain MeasurementsFrequency-Domain Measurements
• Propagation of photon density waves (PDW): PDW = 9 cm, cPDW = 0.06 c (*
• Measure PDW modulation (or amplitude) and phase delay
• RF equipment (100MHz-1GHz)
• Wave strongly damped, challenging measurement
t
I
t0
t
I
t0
t
I
t0Photon density waves
t
I
t0
Phase
Modulation
(* f = 200 MHz, μa = 0.1 cm-1, μs = 10 cm-1 n = 1.37
BMI2 SS07 – Class 2 “DOT Theory” Slide 27
Theoretical Descriptions of NIR Propagation Through TissueTheoretical Descriptions of NIR Propagation Through Tissue
• Quantum Electrodynamics
• Classical Electrodynamics (Maxwell’s equations)
• Radiation Transport Equation
• Diffusion Equation
– Assumes (among other things) that μs(r) >> μa(r).
4
, ,1, , , , , ,
source streaming absorption and
scattering out
, ,
scattering in
a s
s
tS t t t
c t
f t d
rr r r r r
r r
¶
¶
,1, , ,a
tD t t S t
c t
rr r r r r
¶
¶
BMI2 SS07 – Class 2 “DOT Theory” Slide 28
Making the Problem Tractable — Perturbation StrategyMaking 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)| << μa
0(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.,
0 0 0 0
0 0 0 0
0 3
, ,
0 3
, ,
,
,
a a
a a
a aD DV
a a DV
a a DD D
d d d d D D d r
d d d W W D d r
W d W d D
¶ ¶ ¶ ¶
¶ ¶ ¶ ¶
BMI2 SS07 – Class 2 “DOT Theory” Slide 29
Making the Problem Tractable — Perturbation Strategy IIMaking 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:
• Perturbation equations for all source/detector combinations are combined into a system of linear equations, or matrix equation.
0 0 0 0
1
, ,,
a a
Nn n n n
a a Dn
na a n D nD D
d W W D
W d V W d D V¶ ¶ ¶ ¶
BMI2 SS07 – Class 2 “DOT Theory” Slide 30
Making the Problem Tractable — Perturbation Strategy IIIMaking the Problem Tractable — Perturbation Strategy III
• Perturbation equations for all source/detector combinations are combined into a system of linear equations, or matrix equation:
aa D
μd W W
D
1
2
11 12 1 11 12 11
21 22 2 21 22 22
1
1 2 1 2 2
a
aN N
a a a D D DNN Naa a a D D D
M M MN M M MNM a a a D D D
N
d W W W W W Wd W W W W W W
Dd W W W W W W D
D
BMI2 SS07 – Class 2 “DOT Theory” Slide 31
Many different combinations of μa and μs are consistent with any given non-invasive light intensity measurement
2 4 6 8 10
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1 -7
-6
-5
-4
-3
-2
-1
μs
μa
log10(Intensity)
Dilemma:Dilemma:
BMI2 SS07 – Class 2 “DOT Theory” Slide 32
Few spatial distributions of μa and μs are consistent with many nearly simultaneous non-invasive light intensity measurement
(Cavernous hemangioma)
Solution, Part 1:Solution, Part 1:
BMI2 SS07 – Class 2 “DOT Theory” Slide 33
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 measure-ments around its border, is very difficult
Figuring out the difference between the spatial distribu-tions of μa and μs in these two media is much easier
-
=
Solution, Part 2:Solution, Part 2:
BMI2 SS07 – Class 2 “DOT Theory” Slide 34
As a practical matter, most useful method is to use a spatially homogeneous second medium (i.e., reference medium)
µs = 9 cm-1
µa = 0.05 cm-1
µs = 9 cm-1
µa = 0.07 cm-1
µs = 11 cm-1
µa = 0.05 cm-1
µs = 11 cm-1
µa = 0.07 cm-1
µs = 10 cm-1
µa = 0.06 cm-1
Δµs = -1 cm-1
Δµa = -0.01 cm-1
Δµs = -1 cm-1
Δµa = 0.01 cm-1
Δµs = 1 cm-1
Δµa = -0.01 cm-1
Δµs = 1 cm-1
Δµa = 0.01 cm-1
Solution, Part 2:Solution, Part 2:
BMI2 SS07 – Class 2 “DOT Theory” Slide 35
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”)
Solution, Part 3:Solution, Part 3:
BMI2 SS07 – Class 2 “DOT Theory” Slide 36
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
Solution, Part 3:Solution, Part 3:
BMI2 SS07 – Class 2 “DOT Theory” Slide 37
Linear perturbation strategy for image reconstruction
Repeat process just described, for all source-detector combinations.
WEIGHT matrix
Solution, Part 3:Solution, Part 3:
BMI2 SS07 – Class 2 “DOT Theory” Slide 38
Measurement perturbation (difference) is directly proportional to interior optical coefficient perturbation. Weight matrix gives us the proportionality.
µs = 9 cm-1
µa = 0.05 cm-1
µs = 9 cm-1
µa = 0.07 cm-1
µs = 11 cm-1
µa = 0.05 cm-1
µs = 11 cm-1
µa = 0.07 cm-1
µs = 10 cm-1
µa = 0.06 cm-1
Δµs = -1 cm-1
Δµa = -0.01 cm-1
Δµs = -1 cm-1
Δµa = 0.01 cm-1
Δµs = 1 cm-1
Δµa = -0.01 cm-1
Δµs = 1 cm-1
Δµa = 0.01 cm-1
R W X
Linear perturbation strategy for image reconstruction
Solution, Part 3:Solution, Part 3:
BMI2 SS07 – Class 2 “DOT Theory” Slide 39
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
Solution, Part 3:Solution, Part 3:
Linear perturbation strategy for image reconstruction
BMI2 SS07 – Class 2 “DOT Theory” Slide 40
Coping with noise (random error) in clinical measurement data
Linear system solutions are additive:
1 1 11 2 1 2
1 2
W R R W R W R
X X
1 1 1
E
W R W R W
XNoise in data
Noise image
Real-world Issue 1:Real-world Issue 1:
BMI2 SS07 – Class 2 “DOT Theory” Slide 41
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.
Real-world Issue 1:Real-world Issue 1:
Coping with noise (random error) in clinical measurement data
BMI2 SS07 – Class 2 “DOT Theory” Slide 42
,
,
, ; , m ns da m n
ax y
Rw s d x y
¶
¶
-1 -110 cm , 0.06 cm
1/ 3 0.0331 cm
s a
s aD
BMI2 SS07 – Class 2 “DOT Theory” Slide 57
,
,
, ; , m ns dD m n
x y
Rw s d x y
D
¶
¶
, , ,
, , ,
, ; , m n m ns d s d x ys m n
s x y x y s x y
R R Dw s d x y
D
¶ ¶ ¶
¶ ¶ ¶