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Biomedical Imaging 2. Class 2 – Diffuse Optical Tomography (DOT) 01/23/07. Acknowledgment. Dr. Ronald Xu Assistant Professor Biomedical Engineering Center Ohio State University Columbus, Ohio.  PowerPoint PPT Presentation
Transcript of Biomedical Imaging 2
Biomedical Imaging IBiomedical Imaging 2
01/23/07
Acknowledgment
Columbus, Ohio
Slides 11, 1418, 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.ohiostate.edu/resources/medimage_ws2005_Xuimage_workshop_2.16.05.ppt
What Are We Measuring?
Transfer function: T(ri,Ωi)
What Are We Measuring?
Transfer function: T(ri,Ωi)
= T[x(ri[,Ωi])]
More on Transfer 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.
BMI2 SS07 – Class 2 “DOT Theory” Slide *
When Can We Solve for x(r)?
Most generally, T is influenced by x
Most tractable case: T is mediumindependent
i.e., T(x) = T0·x, or T(x) = T0·f(x).
Also sometimes doable: T is not mediumindependent, 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 *
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 *
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 *
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 *
An Intractable Case
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?)
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.
BMI2 SS07 – Class 2 “DOT Theory” Slide *
How Photons Interact with Biological Tissue
s
s
BMI2 SS07 – Class 2 “DOT Theory” Slide *
[From: J. W. Pickering, S. A. Prahl, et al., “Doubleintegratingsphere system for measuring the optical properties of tissue,” Applied Optics 32(4), 399410 (1993).]
Detector
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
1. Inner surfaces are coated with a bright, white, highly reflective material (very high µs, very low µa)
2. Eventually, all nonabsorbed photons are captured by one or another of the detectors
BMI2 SS07 – Class 2 “DOT Theory” Slide *
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
Detector
1. Inner surfaces are coated with a dark, matte, highly absorptive material (very high µa, very low µs)
BMI2 SS07 – Class 2 “DOT Theory” Slide *
Scattering is Caused by Tissue Ultrastructure
(http://omlc.ogi.edu)
Absorption is Caused by Multiple Chromophores
BMI2 SS07 – Class 2 “DOT Theory” Slide *
In NIR Region, Hb and HbO are Major Sensitive Absorber
extinct coeff (cm1/mol/liter)
What Near Infrared Light Can Measure?
Absorption measurement
…
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, …
Noninvasive, real time, local measurement of tissue O2 and HbT is not commercially available
BMI2 SS07 – Class 2 “DOT Theory” Slide *
Why do we want to know μa?
μa = μa(Hboxy) + μa(Hbdeoxy) + μa(H2O) + μa(lipid) + μa(cytoxidase) + μa(myoglobin) + …
μa(X, λ) = ε(X,λ)[X]
Molar extinction coefficient
BMI2 SS07 – Class 2 “DOT Theory” Slide *
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(Hboxy) + μa(Hbdeoxy) + μa(H2O) + μa(lipid) + μa(cytoxidase) + μa(myoglobin) + …
Why do we want to know μa?
BMI2 SS07 – Class 2 “DOT Theory” Slide *
Why Near Infrared? Pros and Cons Compared with Other Imaging Modalities
Advantages:
Noninvasive
Nonradioactive
Hard to quantify
Find embedded tissue heterogeneity
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 SD pairs
High “frame rates” possible
Example: Optical brain imaging
23 cm
TimeResolved Measurements
Measuring the arrival time/temporal spread of short pulses (<ns) due to scattering & absorption (narrowing the “banana”)
Expensive, delicate hardware (singlephoton counters, fast lasers, optical reflections, delays…)
Long acquisition times (low frame rates)
Potentially better spatial resolution than DC measurements
Prompt or ballistic Photons (t = d/c)
d
FrequencyDomain Measurements
Propagation of photon density waves (PDW): PDW = 9 cm, cPDW = 0.06 c (*
Measure PDW modulation (or amplitude) and phase delay
RF equipment (100MHz1GHz)
Photon density waves
(* f = 200 MHz, μa = 0.1 cm1, μs = 10 cm1 n = 1.37
t
I
t0
t
I
t0
t
I
t0
t
I
t0
Phase
Modulation
Quantum Electrodynamics
Radiation Transport Equation
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.,
BMI2 SS07 – Class 2 “DOT Theory” Slide *
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:
Perturbation equations for all source/detector combinations are combined into a system of linear equations, or matrix equation.
BMI2 SS07 – Class 2 “DOT Theory” Slide *
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:
BMI2 SS07 – Class 2 “DOT Theory” Slide *
Dilemma:
Many different combinations of μa and μs are consistent with any given noninvasive light intensity measurement
μs
μa
log10(Intensity)
Solution, Part 1:
Few spatial distributions of μa and μs are consistent with many nearly simultaneous noninvasive light intensity measurement
(Cavernous hemangioma)
Solution, Part 2:
Simplify mathematical problem by introducing an additional lightscattering 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

=
Solution, Part 2:
As a practical matter, most useful method is to use a spatially homogeneous second medium (i.e., reference medium)
µs = 10 cm1
µa = 0.06 cm1
µs = 9 cm1
µa = 0.05 cm1
µs = 9 cm1
µa = 0.07 cm1
µs = 11 cm1
µa = 0.05 cm1
µs = 11 cm1
µa = 0.07 cm1
Δµs = 1 cm1
Δµa = 0.01 cm1
Δµs = 1 cm1
Δµa = 0.01 cm1
Δµs = 1 cm1
Δµa = 0.01 cm1
Δµs = 1 cm1
Δµa = 0.01 cm1
Solution, Part 3:
µs = 10 cm1
µa = 0.06 cm1
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 SS07 – Class 2 “DOT Theory” Slide *
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 SS07 – Class 2 “DOT Theory” Slide *
Solution, Part 3:
Repeat process just described, for all sourcedetector combinations.
WEIGHT matrix
Solution, Part 3:
Linear perturbation strategy for image reconstruction
µs = 9 cm1
µa = 0.05 cm1
µs = 9 cm1
µa = 0.07 cm1
µs = 11 cm1
µa = 0.05 cm1
µs = 11 cm1
µa = 0.07 cm1
Δµs = 1 cm1
Δµa = 0.01 cm1
Δµs = 1 cm1
Δµa = 0.01 cm1
Δµs = 1 cm1
Δµa = 0.01 cm1
Δµs = 1 cm1
Δµa = 0.01 cm1
µs = 10 cm1
µa = 0.06 cm1
Solution, Part 3:
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.
Linear perturbation strategy for image reconstruction
BMI2 SS07 – Class 2 “DOT Theory” Slide *
Realworld Issue 1:
Linear system solutions are additive:
Noise in data
Realworld Issue 1:
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.
Coping with noise (random error) in clinical measurement data
BMI2 SS07 – Class 2 “DOT Theory” Slide *
BMI2 SS07 – Class 2 “DOT Theory” Slide *
BMI2 SS07 – Class 2 “DOT Theory” Slide *
BMI2 SS07 – Class 2 “DOT Theory” Slide *
BMI2 SS07 – Class 2 “DOT Theory” Slide *
BMI2 SS07 – Class 2 “DOT Theory” Slide *
BMI2 SS07 – Class 2 “DOT Theory” Slide *
BMI2 SS07 – Class 2 “DOT Theory” Slide *
BMI2 SS07 – Class 2 “DOT Theory” Slide *
BMI2 SS07 – Class 2 “DOT Theory” Slide *
BMI2 SS07 – Class 2 “DOT Theory” Slide *
BMI2 SS07 – Class 2 “DOT Theory” Slide *
BMI2 SS07 – Class 2 “DOT Theory” Slide *
BMI2 SS07 – Class 2 “DOT Theory” Slide *
BMI2 SS07 – Class 2 “DOT Theory” Slide *
BMI2 SS07 – Class 2 “DOT Theory” Slide *
BMI2 SS07 – Class 2 “DOT Theory” Slide *
BMI2 SS07 – Class 2 “DOT Theory” Slide *
BMI2 SS07 – Class 2 “DOT Theory” Slide *
BMI2 SS07 – Class 2 “DOT Theory” Slide *
BMI2 SS07 – Class 2 “DOT Theory” Slide *
(8 cm)(10.06 cm1) = 80.48
01/23/07
Acknowledgment
Columbus, Ohio
Slides 11, 1418, 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.ohiostate.edu/resources/medimage_ws2005_Xuimage_workshop_2.16.05.ppt
What Are We Measuring?
Transfer function: T(ri,Ωi)
What Are We Measuring?
Transfer function: T(ri,Ωi)
= T[x(ri[,Ωi])]
More on Transfer 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.
BMI2 SS07 – Class 2 “DOT Theory” Slide *
When Can We Solve for x(r)?
Most generally, T is influenced by x
Most tractable case: T is mediumindependent
i.e., T(x) = T0·x, or T(x) = T0·f(x).
Also sometimes doable: T is not mediumindependent, 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 *
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 *
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 *
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 *
An Intractable Case
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?)
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.
BMI2 SS07 – Class 2 “DOT Theory” Slide *
How Photons Interact with Biological Tissue
s
s
BMI2 SS07 – Class 2 “DOT Theory” Slide *
[From: J. W. Pickering, S. A. Prahl, et al., “Doubleintegratingsphere system for measuring the optical properties of tissue,” Applied Optics 32(4), 399410 (1993).]
Detector
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
1. Inner surfaces are coated with a bright, white, highly reflective material (very high µs, very low µa)
2. Eventually, all nonabsorbed photons are captured by one or another of the detectors
BMI2 SS07 – Class 2 “DOT Theory” Slide *
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
Detector
1. Inner surfaces are coated with a dark, matte, highly absorptive material (very high µa, very low µs)
BMI2 SS07 – Class 2 “DOT Theory” Slide *
Scattering is Caused by Tissue Ultrastructure
(http://omlc.ogi.edu)
Absorption is Caused by Multiple Chromophores
BMI2 SS07 – Class 2 “DOT Theory” Slide *
In NIR Region, Hb and HbO are Major Sensitive Absorber
extinct coeff (cm1/mol/liter)
What Near Infrared Light Can Measure?
Absorption measurement
…
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, …
Noninvasive, real time, local measurement of tissue O2 and HbT is not commercially available
BMI2 SS07 – Class 2 “DOT Theory” Slide *
Why do we want to know μa?
μa = μa(Hboxy) + μa(Hbdeoxy) + μa(H2O) + μa(lipid) + μa(cytoxidase) + μa(myoglobin) + …
μa(X, λ) = ε(X,λ)[X]
Molar extinction coefficient
BMI2 SS07 – Class 2 “DOT Theory” Slide *
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(Hboxy) + μa(Hbdeoxy) + μa(H2O) + μa(lipid) + μa(cytoxidase) + μa(myoglobin) + …
Why do we want to know μa?
BMI2 SS07 – Class 2 “DOT Theory” Slide *
Why Near Infrared? Pros and Cons Compared with Other Imaging Modalities
Advantages:
Noninvasive
Nonradioactive
Hard to quantify
Find embedded tissue heterogeneity
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 SD pairs
High “frame rates” possible
Example: Optical brain imaging
23 cm
TimeResolved Measurements
Measuring the arrival time/temporal spread of short pulses (<ns) due to scattering & absorption (narrowing the “banana”)
Expensive, delicate hardware (singlephoton counters, fast lasers, optical reflections, delays…)
Long acquisition times (low frame rates)
Potentially better spatial resolution than DC measurements
Prompt or ballistic Photons (t = d/c)
d
FrequencyDomain Measurements
Propagation of photon density waves (PDW): PDW = 9 cm, cPDW = 0.06 c (*
Measure PDW modulation (or amplitude) and phase delay
RF equipment (100MHz1GHz)
Photon density waves
(* f = 200 MHz, μa = 0.1 cm1, μs = 10 cm1 n = 1.37
t
I
t0
t
I
t0
t
I
t0
t
I
t0
Phase
Modulation
Quantum Electrodynamics
Radiation Transport Equation
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.,
BMI2 SS07 – Class 2 “DOT Theory” Slide *
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:
Perturbation equations for all source/detector combinations are combined into a system of linear equations, or matrix equation.
BMI2 SS07 – Class 2 “DOT Theory” Slide *
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:
BMI2 SS07 – Class 2 “DOT Theory” Slide *
Dilemma:
Many different combinations of μa and μs are consistent with any given noninvasive light intensity measurement
μs
μa
log10(Intensity)
Solution, Part 1:
Few spatial distributions of μa and μs are consistent with many nearly simultaneous noninvasive light intensity measurement
(Cavernous hemangioma)
Solution, Part 2:
Simplify mathematical problem by introducing an additional lightscattering 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

=
Solution, Part 2:
As a practical matter, most useful method is to use a spatially homogeneous second medium (i.e., reference medium)
µs = 10 cm1
µa = 0.06 cm1
µs = 9 cm1
µa = 0.05 cm1
µs = 9 cm1
µa = 0.07 cm1
µs = 11 cm1
µa = 0.05 cm1
µs = 11 cm1
µa = 0.07 cm1
Δµs = 1 cm1
Δµa = 0.01 cm1
Δµs = 1 cm1
Δµa = 0.01 cm1
Δµs = 1 cm1
Δµa = 0.01 cm1
Δµs = 1 cm1
Δµa = 0.01 cm1
Solution, Part 3:
µs = 10 cm1
µa = 0.06 cm1
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 SS07 – Class 2 “DOT Theory” Slide *
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 SS07 – Class 2 “DOT Theory” Slide *
Solution, Part 3:
Repeat process just described, for all sourcedetector combinations.
WEIGHT matrix
Solution, Part 3:
Linear perturbation strategy for image reconstruction
µs = 9 cm1
µa = 0.05 cm1
µs = 9 cm1
µa = 0.07 cm1
µs = 11 cm1
µa = 0.05 cm1
µs = 11 cm1
µa = 0.07 cm1
Δµs = 1 cm1
Δµa = 0.01 cm1
Δµs = 1 cm1
Δµa = 0.01 cm1
Δµs = 1 cm1
Δµa = 0.01 cm1
Δµs = 1 cm1
Δµa = 0.01 cm1
µs = 10 cm1
µa = 0.06 cm1
Solution, Part 3:
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.
Linear perturbation strategy for image reconstruction
BMI2 SS07 – Class 2 “DOT Theory” Slide *
Realworld Issue 1:
Linear system solutions are additive:
Noise in data
Realworld Issue 1:
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.
Coping with noise (random error) in clinical measurement data
BMI2 SS07 – Class 2 “DOT Theory” Slide *
BMI2 SS07 – Class 2 “DOT Theory” Slide *
BMI2 SS07 – Class 2 “DOT Theory” Slide *
BMI2 SS07 – Class 2 “DOT Theory” Slide *
BMI2 SS07 – Class 2 “DOT Theory” Slide *
BMI2 SS07 – Class 2 “DOT Theory” Slide *
BMI2 SS07 – Class 2 “DOT Theory” Slide *
BMI2 SS07 – Class 2 “DOT Theory” Slide *
BMI2 SS07 – Class 2 “DOT Theory” Slide *
BMI2 SS07 – Class 2 “DOT Theory” Slide *
BMI2 SS07 – Class 2 “DOT Theory” Slide *
BMI2 SS07 – Class 2 “DOT Theory” Slide *
BMI2 SS07 – Class 2 “DOT Theory” Slide *
BMI2 SS07 – Class 2 “DOT Theory” Slide *
BMI2 SS07 – Class 2 “DOT Theory” Slide *
BMI2 SS07 – Class 2 “DOT Theory” Slide *
BMI2 SS07 – Class 2 “DOT Theory” Slide *
BMI2 SS07 – Class 2 “DOT Theory” Slide *
BMI2 SS07 – Class 2 “DOT Theory” Slide *
BMI2 SS07 – Class 2 “DOT Theory” Slide *
BMI2 SS07 – Class 2 “DOT Theory” Slide *
(8 cm)(10.06 cm1) = 80.48