Medical Physics Nuclear Medicine Tomographic Imaging Techniques


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Medical Physics Nuclear Medicine Tomographic Imaging Techniques

  1. 1. 1 Medical Physics Nuclear Medicine Tomographic Imaging Techniques Dr Roger Fulton Department of PET & Nuclear Medicine Royal Prince Alfred Hospital Sydney Email: Lectures: Nuclear Medicine Tomography - “Emission Tomography” Planar nuclear medicine images are 2-D projections of a 3-D source distribution. Internal structures are obscured by overlying and uderlying structures. Different projection views can assist, but the person interpreting must mentally assimilate the 3-D distribution. Deep structures may be obscured in all projection views. The alternative is tomographic imaging. Planar Images Ant Post
  2. 2. 2 Tomographic Image A 2-D representation of structures lying within a selected plane or depth in a 3-D object. In nuclear medicine tomography the tomographic image depicts the source distribution, i.e. the distribution of radiotracer in a “slice” of tissue. Medical tomographic systems such as x-ray computed tomography (CT), positron emission tomography (PET), and single photon emission computed tomography (SPECT) reconstruct images of selected planes within the object from projection data obtained at many angles around the object. The reconstruction of slices from the projection data is performed by a mathematical algorithm. Reconstruction is possible provided enough projections are acquired over an adequate range of projection angles. Reconstruction from Projections – “Computed Tomography” 1917 – Initially conceived by Radon, who was concerned with applications in gravitational theory. 1956 – Radioastronomers Bracewell and Riddle were the first to use Radon’s theory. They used it to reconstruct microwave emissions from the sun. 1973 – Hounsfield was the first to use Radon’s theory in medical imaging – first x-ray CT scanner (“CAT” Scanner). Ú • •- = '),(),'( dyyxfxp q
  3. 3. 3 Backprojection Linear Superposition of Back Projections (LSBP). ˜ ¯ ˆ Á Ë Ê ƒ= r 1 imageTrueimageLSBP where r is the distance from the source. Blurring factor ˜ ¯ ˆ Á Ë Ê ƒ= r 1 imageTrueimageLSBP Taking Fourier transform of both sides, .ximage)(LSBPimage)(True frequency,spatialiswhere, 11 Since 1 image)(LSBP image)(True n n n FF F F FF = =˜ ¯ ˆ Á Ë Ê ˜ ¯ ˆ Á Ë Ê = r r (Convolution in the spatial domain Multiplication in the frequency domain).≡ ˜ ¯ ˆ Á Ë Ê = r 1 ximage)(Trueimage)(LSBP FFF
  4. 4. 4 Fourier Transform nximage)(LSBPimage)(True FF = Correcting for 1/r Blurring Taking the inverse Fourier transform of both sides, gƒ= imageLSBPimageTrue Where is the function in the spatial domain whose Fourier transform is equal to n in the frequency domain. Ramp filter Correction Filters † Trueimage = LSBP ƒ g † g
  5. 5. 5 Ramp Filter Since the correction filter has a simple ramp shape in frequency space, it is often called the “ramp” filter. It can be seen that the 1/r blurring is compensated by giving increasing weight to higher spatial frequencies in the Fourier transform of the LSBP image that are needed to represent fine detail. Intuitively, this makes sense since blurring suppresses high spatial frequency information. Hence blurring ought to be reduced by selectively amplifying the higher frequencies. Maximum spatial frequency present in image. Filtered Backprojection (FBP) Apply filter in frequency space by taking the Fourier transforms of the scan profiles (projections) and multiplying each frequency component by a factor proportional to the spatial frequency, or Convolve with the scan profiles in the spatial domain prior to backprojection † g
  6. 6. 6 Filtered Backprojection In practice, instead of the pure ramp filter, a modified ramp filter is used that has a somewhat rounded shape in the frequency domain. This (i) avoids artifacts caused by sharp spatial frequency cut off, and (ii) avoids excessive enhancement of high frequency noise in the image. Filtered backprojection is the most commonly used ECT reconstruction method. Other reconstruction methods will be discussed later. Sampling The projection is not continuous but consists of discrete point by point samples. The Sampling Theorem tells us that in order to recover spatial frequencies in a signal up to the maximum frequency, nmax, the linear sampling distance, d, must be )2( 1 maxn £d i.e. the highest spatial frequency component to be recovered from the data must be sampled at least twice per cycle.
  7. 7. 7 Angular Sampling The angular sampling interval should provide sampling around the periphery at approximately the same intervals as the linear sampling distance, i.e. .projectioneach acrossdistancesamplingnearlitheisand viewoffieldtheofdiametertheiswhere , 2 ewsangular viofNumber d D d Dp = FORMATION OF PROJECTIONS Single Photon Emission Computed Tomography (SPECT) Rotating gamma camera
  8. 8. 8 Triple Head SPECT Scanner Heads rotate 120 degrees to obtain 360 degree angular sampling. Rotation is usually “step-and-shoot” although acquisition during continuous rotation is also possible on some systems. Angular increment typically 3 degrees. Imaging time at each angular position is typically 20 – 40 s. SPECT Sampled volume FOV N x N projection images -> N transaxial slices N typically 64, 128. Axis of rotation
  9. 9. 9 FORMATION OF PROJECTIONS Positron Emission Tomography (PET) b+ g g The positron travels a short distance in tissue, losing energy by Coulomb interactions with electrons, before combining with an electron to form positronium. Positronium has a very short half-life (10-7 s) and annihilates almost immediately. Mass is converted to energy in the form of 2 photons (511 keV each) according to E = mc2 Photons are emitted in opposite directions to conserve momentum. Any residual momentum of positron results in deviation from 180° Positron Decay Coincidence Detection Two detectors oriented at 180 degrees to each other are used to detect annhilation photons. A coincidence circuit detects events that occur within a narrow time interval, typically 12 ns. Coincidence?
  10. 10. 10 PET Spatial Resolution The resolution ultimately obtainable by detection of annihilation photons is limited by two factors. 1.The non-collinearity of the annihilation photons. The deviation from 180 deg is due to the fact that the positron-electron pair is not completely at rest when annihilation occurs. The angular spread of this deviation has a Gaussian distribution with a FWHM of 0.3 degrees. This translates into a resolution effect of about 2.8mm (FWHM) for detectors 100cm apart. 2. The range of the positrons prior to annihilation. This contributes a resolution effect of up to ~3mm (FWHM). The combination of these 2 effects imposes a resolution limit of 1.5 – 3.0mm (FWHM) in PET. The reconstructed spatial resolution of a typical PET scanner is ~ 4.5mm (FWHM). PET Spatial Resolution PET III 1975 ECAT II 1977 NeuroECAT 1978 ECAT 931 1985 ECAT EXACT HR+ 1995
  11. 11. 11 PET 8 x 8 crystal elements 8 transaxial slices Detectors in coincidence Block detector Full ring PET scanner PET "Block" Detector Scintillator array PMTs Histogram X = A + B - C - D A + B + C + D Y = A - B + C - D A + B + C + D B D A X Y
  12. 12. 12 PET Scanner Construction PET vs SPECT Acquisition PET has electronic collimation.
  13. 13. 13 2-D PET For N crystal rings we have (2N-1) transaxial reconstruction planes. N direct planes (N-1) cross planes Coincidence Event Types True Coincidence Random/Accidental Coincidence Multiple Coincidences Scattered Coincidence ¸ ˚ ˚ ˚ Not all coincidence events are desirable. Only true unscattered coincidences contribute useful data for image formation
  14. 14. 14 PET Sinogram Lines of Response (LORs) PET Sinogram Position Projection angle Sinogram Reconstructed slice
  15. 15. 15 Whole Body PET Data acquired as bed moves in discrete increments under computer control Emission & transmission acquisitions interleaved or simultaneous Axial FOV overlap Bed translation Normal scan 18FDG PET
  16. 16. 16 3D Acquisition 2D 3D Removing the interplane septa and accepting coincidences between any pair of detector rings provides a big increase in count rate sensitivity (factor of 3 to 5). This septa-less acquisition is referred to as “3D” PET. (Note: Reconstruction yields a 3D reconstruction volume in both 2D and 3D PET). Benefits are greater sensitivity, leading to shorter acquisition times or ability to use smaller doses of radiotracer. Disadvantages: much larger data sets, reconstruction algorithm more complicated, longer reconstruction times. 2D Projection is a 1-D collection of parallel ray sums originating from a 2D slice 3D Projection is a 2-D collection of parallel ray sums originating from a 3D object 2D PET vs 3D PET Rays confined to a single plane. Reconstruct slice by slice. Rays intersect multiple reconstruction planes. Require “fully 3D” reconstruction algorithm. Ref: Bendriem and Townsend, 1998
  17. 17. 17 Reconstruction Methods • Filtered Backprojection (FBP) • Iterative Methods – Simple Iterative Reconstruction Technique (SIRT) • Algebraic Methods - Algebraic Reconstruction Technique (ART) • Iterative Least Squares Methods (ILST) • Direct Analytical Methods e.g. Fourier Reconstruction • Iterative Statistical Methods e.g. Maximum Likelihood Expectation Maximisation (ML-EM) Fourier Reconstruction The 2D function f(x,y) representing the unknown radiotracer distribution can be expressed as a sum of sine and cosine waves propagating in various directions across the plane. The amplitudes of these waves are denoted by the Fourier coefficients F(u,v). Fourier reconstruction relies on the fact that the amplitudes of the waves propagating at angle q are equal to the Fourier coefficients of the projection P(k,q) at the same angle, i.e. ( ) .andtanwhere,),(),( 221 vukuvkPvuF +±=== - qq Central Slice Theorem
  18. 18. 18 Fourier Reconstruction (continued) By taking projections at several angles,and then taking their Fourier transforms, values of F(u,v) can be determined along radial lines as shown at right. The dots represent the location of estimates of the object’s Fourier transform. If sufficient projections are available, the original object can be reconstructed by taking the inverse Fourier transform. In practice it is difficult to calculate the inverse Fourier transform from radially sampled data, and the transform is usually first interpolated onto a rectangular grid. The interpolation must be very accurate, especially at large radii where sampling is relatively sparse, and suitable interpolation algorithms are very expensive computationally. Interpolate to rectangular grid Inverse Fourier Transform Reconstruction Iterative Methods Iterative reconstruction methods attempt to progressively improve the estimate of f(x,y) rather than estimate it by direct inversion of the image transform. Such methods allow more realistic modelling of the detection process than the simple line integral we had in filtered backprojection, Ú • •- = '),(),'( dyyxfxp q The ability to incorporate accurate models for noise, and other effects such as attenuation and scatter, can be a significant advantage. Iterative methods also have the ability to incorporate a priori information about the image in the reconstruction process. Such information can be used to help ensure an acceptable reconstruction. For example, since f(x,y) represents tracer concentration, it is reasonable to require that successive image estimates are non-negative in every pixel.
  19. 19. 19 Iterative Algorithm Requirements 1. A representation of the object to be reconstructed as a finite set of volume elements (voxels). 2. A model of the detection process that specifies the relationship between the observed projection data and the image. 3. A noise model to define the type of distribution of data elements about their expectation value. 4. An objective function that provides a measure of the goodness-of-fit of the image to the measured data and applies any a priori constraints. 5. An iterative algorithm that can successively improve the image estimate by converging to a solution that maximizes the objective function.
  20. 20. 20 The Model of the the Detection Process Let m be the number of projection values, given by the product of the number of projection bins Nb at each angle and the number of projection angles Nq. We represent the m projection values as p={pj, j=1,2,..,m} where pj is the number of photons recorded in the jth projection bin. We represent the n voxel values in the image as f = {fi, i=1,n}. The required model can then be written p = Af, where A is the mxn transition matrix that gives the probability of a photon emitted from voxel i being detected in projection bin j. This probability can take into account factors affecting projection measurements such as projection geometry, detector effciency and location, resolution, attenuation, scatter and so on. Hence iterative reconstruction techniques can accurately model the detection process and thus correct for physical effects involved in the detection process.  Â ˛ ˝ ¸ Ó Ì Ï -˜ ¯ ˆ Á Ë Ê = j i iji i ijij fAfApL ln)(f    = = = + = n i k iji j m j jim j ji k ik i fA p A A f f 1 1 1 1 Maximum Likelihood Expecation Maximisation (ML-EM) EM AlgorithmObjective Function Seeks the most likely solution consistent with the measured projections, i.e. finds the image f that is most likely to have produced the projections p. Lang K and Carson R, “EM reconstruction agorithms for emission and transmission tomography”, J Comput Assist Tomogr, 8(2):306-316, 1984. ML-EM Algorithm Accounts for noise much better than FBP but computationally demanding.
  21. 21. 21 Ordered Subsets EM Divide the projections into “subsets” each consisting of 2 or more projections. Apply ML-EM to one subset at a time. Equivalent results to EM but faster. Acceleration factor is approximately equal to number of subsets. PET Summary Cyclotron Radioisotope Production Chemistry Radiotracer Synthesis Physiological process Tissue uptake Injection into blood Radioactive Decay Positron emission PET Camera Acquire projections Computer Tomographic Reconstruction 18FDG PET Brain Scan
  22. 22. 22 18FDG PET Brain Scan coronal sagittal transaxial 18FDG – Regional Metabolic Activity listening seeing
  23. 23. 23 18FDG – Regional Metabolic Activity thinking moving 6-[18F]Fluoro-DOPA in Parkinson’s Disease
  24. 24. 24 microPET 18FDG Rat Study Further reading: Sorenson JA, Phelps ME, “Physics in Nuclear Medicine”, Grune and Stratton, Orlando, 1987. Bendriem B, Townsend DW (ed.), “The Theory and Practice of 3D PET”, Kluwer Academic Publishers, Dordrecht, 1998. Henkin RE, “Nuclear Medicine”, Mosby, St Louis, 1996. Sarper RM, “Nuclear Medicine Instrumentation”, CC Thomas, Springfield, 1984. Murray IPC, Ell P (ed.), “Nuclear Medicine in Clinical Diagnosis and Treatment”, Churchill Livingstone, Edinburgh, 1998. Let’s Play PET Uniserve oactivity