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Altaf H. Khan (Biostatistician)
Division of Biostatistics & Modeling
King Abdullah International Medical
Research Center
National Guard Health Affairs
Riyadh, Saudi Arabia
Introduction
 Challenges
 Imaging Modalities
 Computed Tomography (Reconstruction Algorithms)
 Diagnostic Tools
 Statistical Inferences (Positron Emission Tomography)
 Mathematical and Statistical Challenge and Issues.
 Conclusion
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Non-invasive anatomic and functional imaging ranks one of the foremost advances in medicine over
the past 100 years and promises to continue over the coming years.
Advancement in medical modalities have greatly improved clinicians’ ability to detect, diagnose and
treat disease and injury at a very early stage. And this paved the way, as people have longer life
expectancy and experiencing higher quality of life than ever before.
Utilizing sophisticated mathematical and statistical algorithms to create images of a patient’s internal
anatomy and convert them to film through diagnostic imaging has revolutionized the way many
diseases and injuries are detected, diagnosed, and treated.
Now, millions of people could avoid invasive, and costly diagnostic procedures through the use of
modern imaging technology, without an incision physician can see inside the body, and allows intricate
procedures on fragile organs, without surgery
Source: Ensuring Quality through Appropriate Use of
Diagnostic imaging (AHIP)
Images or volumes
Raw data
Patient/
subject
Imaging Device
Physical property
Preprocessing
Reconstruction
Processed data
Interpretation
Probabilistic Inference
A General schematic illustration of steps in
radiological modalities
Decision
Physician
Source: Dimitri Van De Ville




Mathematical Models
Statistical Models
Bottom Up Approach
Top Down Approach
Starts with the hypotheses of
physical or biological
mechanism
Build the mathematical
equations with the components
involve.
Based upon simulation to
generate predictions
Ordinary, partial differential or
integral equations



Starts with a dataset (often very
large)
Explore the data with statistical
models to see any pattern in the
data.
Make predictions based upon the
structure within the data, e.g.,
i. Regressions analysis
ii. Principal Components analysis
iii. Network analysis
iv. Clustering, etc.
◦ Non-Destructive Testing
◦ Geophysical Exploration
◦ Modern medicine
– Population screening and risk assessment
– Disease diagnosis
– Prognostic prediction
– Monitoring disease progression
– Imaging-guided treatments
– To understand biological systems
– Clinical trials, both as the clinical and surrogate endpoints
The imaging methods used in biomedical applications as follows:
 X-ray projection imaging,
 X-ray computed tomography (CT),
Magnetic resonance imaging (MRI) and magnetic
spectroscopy,
 Single photon emission computed tomography (SPECT),
 Positron emission tomography (PET),
 Ultrasonic,
 Electrical source imaging (EIS),
 Electrical impedance tomography (EIT),
 Magnetic source imaging (MSI), and
 Medical optical imaging.
resonance
Tomography:
The word “τόμσ” (tomos) is Greek word, means a slice, and
“γραϕό” (graphό). Thus tomography is the science of drawing slices. By
stacking these slices to obtain the 2D or 3D objects.
Reconstruction:
In medical imaging the object is reconstructed without dissecting
or cutting the object, The reconstruction is done thru indirect measurement as
some function of interest. Moreover, what needs to be recorded and how it is
reconstructed solely based upon the physical process involved.
X-Ray CT:
The number of x-ray photons transmitted through the patient
along individual projection lines.
Nuclear medicine:
The number of photons emitted from the patient along individual
projection lines.
Ultrasound diffraction tomography:
The amplitude and phase of scattered waves along a particular line
connecting the source and detector.


Direct Analytical Methods:
The most widely used methods of image reconstruction are direct analytical methods
because they are relatively quick, but the problem associated with these images
tend to be ‘streaky’ and display interference between regions of low and high tracer
concentration.
Algorithm Steps:

1. Acquire 1D projections

2. Convert 1D projections to Fourier Transforms

3. Build up Fourier Transforms into 2D image

4. Reconstruct 2D real space image from 2D Fourier Transform image
Iterative Methods:
Images reconstructed from iterative methods are computationally much more intensive;
however, with the development of parallel architectures
and
current
generation PC clusters in the GHz range, the potential for using these in
conjunction with iterative techniques, the problem becomes less awful.
Flow chart describing the various steps in an
iterative reconstruction process.
Source: IAEA Human Health Reports No. 9
Analytic reconstruction methods
(projection - backprojection algorithms)
filtered back-projection
back-projection
Radon J.
On the determination of functions from their integrals
along
certain manifolds [in German].
Math Phys Klass 1917;69:262-277.on filtering

Radon Transform
ˆ (  , ) 
p
L  x cos
pˆ (  , ) 
 


f ( x, y ) ds,
 y sin   
  f ( x, y) (x cos  y sin    )dxdy
  
Geometry
Source: Khan and Chaudhuri , 2014
Composite Materials
Fourier Slice Theorem
1D FT = a slice of 2D FT
Shepp-Logan Phantom and Its Radon Transform
a. Sum of ellipses’ characteristic
functions with heights given by the
color of each ellipse.
b. The Radon transform of a sum of
ellipses’ characteristic functions.
c. Radon transform. We can observe that the Radon transform is
the sum of the four Radon transforms..
Source: Matthieu SIMEONI,
MA-MA2, EPFL. Summer Project:
Statistical Inference in PET

Mathematically, backprojection operation is defined as:
𝜋
𝑓 𝑥, 𝑦 =
𝑝 𝑥𝑐𝑜𝑠𝜃 + 𝑦𝑠𝑖𝑛𝜃, 𝜃 𝑑𝜃
0

Geometrically, the backprojection operation simply propagates the measured sinogram back
into the image space along the projection paths.
The backprojection images of the Shepp-Logan phantom
- Comparison of original
phantom with the
backprojection image
seems to be blurred.
Specifically,
- for a point source at the
origin δ(x,y), the intensity
of the backprojection
image rolls off slowly as
1/r.
- convolve 1/r with f(x,y)
I view
2 views
10 views
FBP image
Original image
30 views
80 views
50 views
Projection
Backprojection
Ramp filter
Filtered sonogram
Sinogram
Analytical Methods

Advantages
◦ Fast
◦ Simple
◦ Predictable, linear behaviour

Disadvantages
◦ Not very flexible
◦ Image formation process is not modelled  image properties are sub-optimal
(noise, resolution)
Iterative Methods
 Choose a number between 1 and 25
Person:1
Estimate
number
Update Estimate
Person:2
Compare with actual
number
<,>=?
Match

discreteness of data when included in the model
 it is easy to model and handle projection noise,
especially the counts are low.
 it is easy to model the imaging physics such as
geometry, non-uniform attenuation, scatter, etc.
quantitative imaging possible
 Can accurately model the image formation process (use with non-standard
geometries, e.g. not all angles measured, gaps)
 Allow use of constraints and a priori information (non-negativity, boundaries)
 Corrections can be included in the reconstruction process (attenuation, scatter, etc)
amplification of noise
 long calculation time, and less predictive behaviour
(noise? convergence?
Example of
beam
hardening
artifacts
A CT image with a cardiac motion
artifact in the ascending aorta that
mimics an aortic dissection.
streaking effect on an image
caused by metal objects
Beam hardening
artifact. The
density
of tissue behind
dense bone
changes because
of beam hardening
artifact.
Motion artifact (3)
Metal artifact
(4)
Scanning principle of helical
CT
Streak artifact. B) Correction of
the artifact by adaptive filtering.
Partial volume artifact
Practical Issues and Artifacts
150 angles
300 angles
Aliasing: insufficient angular sampling
- reducing the number of projections desirable for the purpose of
a. Reducing scanning time
b. Reducing noise
c. Reducing motion artifact
d. Reducing patient dose
600 angles
1200 angles
Aliasing - Insufficient radial sampling
– occurs when there is a sharp intensity change
caused by, for example, bones.
Motion artifact:
caused by patient motion,
such as respiration and heart
beat, during data acquisition
Incomplete /Missing Data:
– portion of data can not be acquired due to physical or instrumental limitations
a. limited angles: Some views can not be acquired due to physical or
instrumental limitations
b. metal artifacts: In CT, metal blocks the radiation, leading to missing
data in its shadow.
Goals of Statistical Inferences and Models
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Summarize data
Estimation: point and interval estimates
Inference: hypotheses / relationships
Power and sample size
Prediction
Schematic overview of PET (Source: Sabrina Dill, 2013)
Illustrations of true (top, left), random
(top, right), scatter (bottom, left) and multiple
(bottom, right) coincidences (EANM)
Goals of Statistical Inferences and Models
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Positron Emission Tomography (PET) Data
Compartmental Analysis
Model Selection
Smoothing Neuroimaging Data in Space
Partial Volume Correction
Functional Principal Component Analysis
Non-Parametric Spatial Modeling with Wavelets
Bayesian Self Similarity
Modeling for the Population
Motivation
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Linear Models = cornerstone of statistical methods - linear regression
to complex models
Imaging data – statistical methods to look for “regional effects” or at
the voxel level
Across subjects: Tissue differences between groups
(e.g., voxel/tensor-based morphometry)
Within subjects: PET (positron emission tomography), fMRI
(functional MRI) – activation in brain due to thought, stimulus or task
Non-brain: Diffusion (DWI, DTI, tractography), Bone mineral density
etc., etc.
Challenges
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Generating suitable statistical imaging models
Dealing with highly multivariate responses (curse of
dimensionality)
Defining imaging “hypotheses”
Creating computationally efficient analysis procedures
Goals of Statistical Inferences and Models (Statistical Modeling
Strategy)
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Propose a model for the data
Fit the model
Assess the model’s adequacy
Fit other plausible models
Compare all fitted models
Interpret the best model
Acknowledgement
I am very grateful to Dr. Mohammed A. Hussein, Chairman of the
Department of Bioinformatics, Biostatistics & Modeling and Data
Management Center at the King Abdullah International Medical Research
Center (KAIMRC) for his continuous encouragement, and his constant kind
help and support. I also want to extend my thanks to Dr. Ahmed Al Askar, the
Director of the KAIMRC, and Dr. Bander Al Kanawy, the Chief Executive
Officer of the National Guard Health Affairs for his approval for the travel
grants.
Moreover, I also want to thank my department colleagues and friends for their
moral support and help.
Many of the slides in this lecture have been adapted
from slides available from various sources, and
especially thanks to those who put interesting material
on the internet.
Thank you!