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OMICS Group International is an amalgamation of Open Access publications and worldwide international science conferences and events. Established in the year 2007 with the sole aim of making the information on Sciences and technology ‘Open Access’, OMICS Group publishes 400 online open access scholarly journals in all aspects of Science, Engineering, Management and Technology journals. OMICS Group has been instrumental in taking the knowledge on Science & technology to the doorsteps of ordinary men and women. Research Scholars, Students, Libraries, Educational Institutions, Research centers and the industry are main stakeholders that benefitted greatly from this knowledge dissemination. OMICS Group also organizes 300 International conferences annually across the globe, where knowledge transfer takes place through debates, round table discussions, poster presentations, workshops, symposia and exhibitions.Open Access publicationsscholarly journalsInternational conferences

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OMICS Group International is a pioneer and leading science event organizer, which publishes around 400 open access journals and conducts over 300 Medical, Clinical, Engineering, Life Sciences, Pharma scientific conferences all over the globe annually with the support of more than 1000 scientific associations and 30,000 editorial board members and 3.5 million followers to its credit. OMICS Group has organized 500 conferences, workshops and national symposiums across the major cities including San Francisco, Las Vegas, San Antonio, Omaha, Orlando, Raleigh, Santa Clara, Chicago, Philadelphia, Baltimore, United Kingdom, Valencia, Dubai, Beijing, Hyderabad, Bengaluru and Mumbai.

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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

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Introduction Challenges Imaging Modalities Computed Tomography (Reconstruction Algorithms) Diagnostic Tools Statistical Inferences (Positron Emission Tomography) Mathematical and Statistical Challenge and Issues. Conclusion 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 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

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Source: Ensuring Quality through Appropriate Use of Diagnostic imaging (AHIP)

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Patient/ subject Imaging Device Preprocessing Reconstruction Interpretation Physician Decision Physical property Raw data Processed data Images or volumes Probabilistic Inference A General schematic illustration of steps in radiological modalities A General schematic illustration of steps in radiological modalities Source: Dimitri Van De Ville

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Mathematical Models Statistical Models Bottom Up 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 Bottom Up 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 Top Down Approach 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. Top Down Approach 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.

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◦ 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 ◦ 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

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The imaging methods used in biomedical applications as follows: X-ray projection imaging, X-ray computed tomography (CT), Magnetic resonance imaging (MRI) and magnetic resonance 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. The imaging methods used in biomedical applications as follows: X-ray projection imaging, X-ray computed tomography (CT), Magnetic resonance imaging (MRI) and magnetic resonance 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.

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Tomography: The word “τόμσ” (tomos) is Greek word, means a slice, and “γρα ϕ ό” (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. Tomography: The word “τόμσ” (tomos) is Greek word, means a slice, and “γρα ϕ ό” (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.

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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. 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.

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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. 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.

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Flow chart describing the various steps in an iterative reconstruction process. Source: IAEA Human Health Reports No. 9

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Radon Transform 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: on filtering 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: on filtering

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Radon TransformGeometry

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Source: Khan and Chaudhuri, 2014 Composite Materials

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Fourier Slice Theorem 1D FT = a slice of 2D FT

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Shepp-Logan Phantom and Its Radon Transform

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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

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Mathematically, backprojection operation is defined as: Geometrically, the backprojection operation simply propagates the measured sinogram back into the image space along the projection paths.

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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)

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Original image FBP image I view 2 views10 views 30 views 50 views 80 views Projection Sinogram Ramp filter Filtered sonogram Backprojection

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Advantages ◦ Fast ◦ Simple ◦ Predictable, linear behaviour Disadvantages ◦ Not very flexible ◦ Image formation process is not modelled image properties are sub-optimal (noise, resolution) Advantages ◦ Fast ◦ Simple ◦ Predictable, linear behaviour Disadvantages ◦ Not very flexible ◦ Image formation process is not modelled image properties are sub-optimal (noise, resolution) Analytical Methods

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Choose a number between 1 and 25 Iterative Methods Person:1Person:2 Estimate number Estimate number Compare with actual number Compare with actual number Update Estimate =? Match

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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? 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?

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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

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Motion artifact (3) Metal artifact (4) Scanning principle of helical CT Beam hardening artifact. The density of tissue behind dense bone changes because of beam hardening artifact. Partial volume artifact Streak artifact. B) Correction of the artifact by adaptive filtering.

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Practical Issues and Artifacts 15 0 angles 30 0 angles 60 0 angles120 0 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

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Aliasing - Insufficient radial sampling – occurs when there is a sharp intensity change caused by, for example, bones. 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. Motion artifact: caused by patient motion, such as respiration and heart beat, during data acquisition

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Summarize data Estimation: point and interval estimates Inference: hypotheses / relationships Power and sample size Prediction Summarize data Estimation: point and interval estimates Inference: hypotheses / relationships Power and sample size Prediction Goals of Statistical Inferences and Models

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Schematic overview of PET (Source: Sabrina Dill, 2013)

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Illustrations of true (top, left), random (top, right), scatter (bottom, left) and multiple (bottom, right) coincidences (EANM)

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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 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 Goals of Statistical Inferences and Models

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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. 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. Motivation

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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 Generating suitable statistical imaging models Dealing with highly multivariate responses (curse of dimensionality) Defining imaging “hypotheses” Creating computationally efficient analysis procedures Challenges

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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 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 Goals of Statistical Inferences and Models (Statistical Modeling Strategy)

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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. 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. Acknowledgement

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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.

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Thank you!

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