Implementing the Probability Matrix Technique for Positron Emission Tomography By: Chris Markson Student Adviser: Dr. Kaufman

Introduction ● PET vs. CAT – PET ~ metabolic – CAT ~ anatomical ● The Process – Tagged chemical compound – Positrons in compound meet electrons in the body~ annihilation – Photons produced at 180° – Detection – Problem: Reconstruct the image using the tube data

● Put grid over organ of interest – 2 dimensions - pixels – 3 dimensions - voxels ● Finer grid, more detail ● Finer grid, more computation Grids time and space

● C(tubes,pixels) = probability that events in a pixel will be detected by a tube ● If 128 detectors, 128 pixels on a side,125 million elements ● About 1% nonzero ● Want to find the emissions x to approximate tube data y such that C x ~ y and x>=0 ● Sum of the elements in x should equal the sum of the elements in y so all annihilations are accounted for ● Use iterative approaches that only require matrix vector multiplication Probability Matrix Approach

● Translate from FORTRAN into C++ and speed up code that generates the C matrix and does matrix by vector multiplication – Take out redundant code for successive subroutine calls – FORTRAN was very unstructured ● Because have more memory, do not have to use circular symmetry – does this lead to faster code? ● If one uses structures and linked lists – does this help? ● Prior research suggested that EM algorithm for maximum likelihood works better from uniform starting guess rather than random – Does starting with solution from coarser grid help? – Should you smooth coarser grid solution? My Responsibilities

● One iteration: – Set s = C T x – Set z = y(t)/s t for t = 1, 2,..., T. – Set w = Cz – Set x b new = x b w b for b = 1, 2,..., B. EM Algorithm

● Images Studied – 10 million vs. 1 million probabilities ● Uniform guess – total tube count divided by number of pixels ● Block expansion – Start with 64x64 data, EM Algorithm, expand to 128x128(1 pixel becomes 4 pixels), old solution becomes initial guess for 2 nd EM run ● Smoothed expansion – Same procedure, (1 pixel becomes 4 pixels)  averaging middle pixels after expansion – Best image produced with tumors standing out Image Output

Tube Data 128x128 ~ 1m 64x64 ~ 1m smoothed 128x128 ~ 10m

uniform10m No expansion, uniform initial guess. block10m 64x64 initial guess into EM, block expansion. wexpand10m 64x64 initial guess into EM, smoothed expansion. **Best result**

uniform1m No expansion, uniform initial guess. block1m 64x64 initial guess into EM, block expansion. wexpand1m 64x64 initial guess into EM, smoothed expansion.

smuniform1m Smoothed first. No expansion, uniform initial guess. smwblockexpand1m Smoothed first. 64x64 initial guess into EM, block expansion. smwexpand1m Smoothed first. 64x64 initial guess into EM, smoothed expansion.

diffexbl1m Difference between smoothed expansion and block expansion smdiffexbl1m Smoothed tube data. Difference between smoothed expansion and block expansion smdiffunex1m Smoothed tube data. Difference between uniform and smoothed expansion.

EM Iteration Analysis

Full Square Code ● Removal of repeating triangle code ● Entire code much slower when compared to the repeating triangle code ● EM algorithm faster – Removal of encode, decode, a1, a2 functions. Easier matrix by vector multiplication ● Creating C matrix takes longer

Times All timings performed on Intel 1.4 GHz Centrino Chip, 512 MB RAM

Linked List ● Theory: Having pointers to first element in each array save on offset calculations used in arrays. ● EM() (16 iterations) – No LL ● Avg processing time: ms – With LL ● Avg processing time: ms – Difference: With LL ms slower

Linked List Times (continued) ● Program Times: – No LL: ● Avg processing time: ms – With LL ● Avg processing time: ms – Difference: With LL ms slower

● Original Code ~ FORTRAN ● Translated and manipulated ~ C++ ● Image manipulation and analysis ~ MATLAB Tools Used